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Diss. ETH No. 19652

AFFINE AND POLYNOMIALPROCESSES

A dissertation submitted to

ETH ZURICH

for the degree of

Doctor Of Sciences

presented by

CHRISTA CUCHIERO

Dipl. Ing. TU Vienna

born 13 April 1983

citizen of Austria

accepted on the recommendation of

Prof. Dr. Josef Teichmann examiner

Prof. Dr. Walter Schachermayer co-examiner

Ass. Prof. Dr. Johannes Muhle-Karbe co-examiner

2011

Abstract

This thesis is devoted to the study of affine and polynomial processes. Both areparticular classes of continuous-time Markov processes, whose specific proper-ties are determined by the following assumptions on the associated semigroups:in the case of affine processes, the Fourier-Laplace transform of the marginaldistributions is supposed to depend in an exponential affine way on the initialstate; in the case of polynomial processes, the semigroup is assumed to mappolynomials into polynomials. This implies in particular that under momentassumptions, affine processes are a subclass of polynomial processes.

Apart from some recent developments in multivariate stochastic volatilitymodeling, affine processes have mainly been studied on the particular statespace Rm

+ × Rn, where a complete characterization was provided by Duffie,Filipovic, and Schachermayer [2003].

Building on the results of affine processes on Rm+ × Rn, the aim of this

thesis is twofold: first, the analysis of affine processes on general state spaces,in particular convex cones and the subclass of symmetric cones; second, thestudy of analytically tractable extensions of the affine class, which leads tothe introduction of polynomial processes.

The first crucial property needed to study affine processes is the differ-entiability of the Fourier-Laplace transform with respect to time, which iscalled regularity. We establish this property for all affine processes on anystate space. Beyond that, we prove that every affine process admits a cadlagversion and is a semimartingale up to its lifetime.

In order to further characterize affine processes, we strengthen our as-sumptions on the state space and consider affine processes on convex conesand in particular on symmetric cones. The latter setting contains the coneof positive semidefinite matrices, which is – in view of multivariate stochas-tic volatility modeling – particularly relevant for practical applications. Onirreducible symmetric cones we then provide a full characterization of affineprocesses, meaning that the parameters of the semimartingale characteristicsnecessarily satisfy some admissibility conditions, which are also sufficient forexistence.

i

ii Abstract

The second line of research, namely a generalization of affine processes,is based on the observation that the expected value of a polynomial of anaffine process is again a polynomial in its initial value. Taking this propertyas definition of a class of Markov processes, which we call polynomial, givesrise to an analytically tractable extension of the affine class, for which thecalculation of moments only requires the computation of matrix exponentials.

The last part of the thesis is devoted to applications of affine and polyno-mial processes, where we study multivariate affine stochastic volatility modelsand variance reduction techniques in Monte-Carlo simulations by exploitingthe properties of polynomial processes.

Kurzfassung

Das Thema dieser Dissertation sind affine und polynomiale Prozesse. Beidesind bestimmte Klassen zeitkontinuierlicher Markov-Prozesse, deren spezifis-che Eigenschaften durch folgende Annahmen an die Halbgruppen bestimmtsind: im Fall der affinen Prozesse hangt die Fourier-Laplace-Transformierteder Randverteilungen exponentiell affin vom Anfangszustand ab, wahrend imFall der polynomialen Prozesse die Halbgruppe Polynome auf Polynome ab-bildet, d.h., der Erwartungswert eines jeden Polynoms ist wieder ein Polynomim Anfangwert. Insbesondere bedeutet dies, dass affine Prozesse unter Mo-mentenbedingungen eine Unterklasse der polynomialen Prozesse bilden.

Abgesehen von neueren Entwicklungen im Bereich multivariater Volatili-tatsmodellierung wurden affine Prozesse hauptsachlich auf dem ZustandsraumRm

+×Rn untersucht und auf diesem von Duffie et al. [2003] vollstandig charak-terisiert.

Aufbauend auf diesen Ergebnissen ist das Ziel dieser Arbeit, einerseitsdie Erforschung affiner Prozesse auf allgemeinen Zustandsraumen, insbeson-dere auf konvexen und symmetrischen Kegeln, und andererseits die Unter-suchung analytisch gut handhabbarer Erweiterungen der affinen Klasse, waszur Einfuhrung polynomialer Prozesse fuhrt.

Die erste entscheidende Eigenschaft fur die Analysis affiner Prozesse istdie Differenzierbarkeit der Fourier-Laplace-Transformierten bezuglich der Zeit,was als Regularitat bezeichnet wird. Wir zeigen diese Eigenschaft fur alleaffinen Prozesse auf jedem beliebigen Zustandsraum. Daruber hinaus beweisenwir, dass jeder affine Prozess eine cadlag Version besitzt und ein Semimartingalbis zum Ende seiner Lebenszeit ist.

Zur weiteren Charakterisierung affiner Prozesse treffen wir starkere An-nahmen an den Zustandsraum und schranken uns auf konvexe und sym-metrische Kegel ein. Letztere enthalten den Kegel der positiv semidefinitenMatrizen, der - in Anbetracht multivariater stochastischer Volatilitatsmodelle- von besonderer Bedeutung fur die Praxis ist. Auf irreduziblen symmetrischenKegeln erhalten wir eine vollstandige Charakterisierung affiner Prozesse, d.h.,die Parameter der Semimartingalcharakteristiken erfullen notwendigerweise

iii

iv Kurzfassung

bestimmte Zulassigkeitsbedingungen, die sich auch als hinreichend in Hinblickauf die Existenz affiner Prozesse erweisen.

Die zweite Forschungsrichtung, namlich eine Verallgemeinerung affiner Pro-zesse, beruht auf der Beobachtung, dass der Erwartungswert eines Polynomseines affinen Prozesses wieder ein Polynom in dessen Anfangswert ist. Definiertman nun polynomiale Prozesse uber diese Eigenschaft, fuhrt dies zu einerneuen Prozessklasse, fur die Momente auf besonders einfache Weise durchMatrixexponentiale berechnet werden konnen.

Der letzte Teil der Arbeit widmet sich Anwendungen von affinen undpolynomialen Prozessen. In diesem Zusammenhang analysieren wir multivari-ate affine stochastische Volatilitatsmodelle und Techniken zur Varianzreduk-tion in Monte-Carlo-Simulationen, wobei wir die Eigenschaften polynomialerProzesse ausnutzen.

Acknowledgment

First of all, I would like to express my gratitude to my advisor, Josef Teich-mann, whose expertise, enthusiasm for research, understanding and encour-agement added considerably to the completion of my thesis. I appreciatenot only his mathematical comprehension and vast knowledge in many ar-eas, but also his kindness and amicable way of mentoring. Without doubthe aroused my joy of mathematical research, inspired me with his invaluableideas, motivated me with fascinating discussions and enabled me to deepenmy understanding of stochastic analysis and mathematical finance.

My special thanks also go to Damir Filipovic, my co-supervisor, who pro-vided me with helpful advice and very in-depth inputs while working in Vi-enna. The collaboration with him considerably enriched my graduate experi-ence and led to a very fruitful scientific work.

I would also like to thank Walter Schachermayer, who was head of theInstitute for Mathematical Methods in Economics and the Research Group forFinancial and Actuarial Mathematics (FAM) at TU Vienna, for the excellentworking environment and pleasant atmosphere at FAM. Beyond that, I amalso very grateful to him for refereeing my thesis. Likewise, I also want tothank Johannes Muhle-Karbe for accepting the task of writing a report formy thesis and providing me with very helpful suggestions.

Furthermore, I would particularly like to express my thanks to my co-authors Martin Keller-Ressel and Eberhard Mayerhofer. Our common workand fruitful discussions constituted a real source of motivation throughout theyears of my graduate studies. Likewise, I also want to thank Elisa Nicolatoand David Skovmand, who opened new doors and perspectives to some appli-cations of our field of research. Our time spent together, at and outside work,has always been a great pleasure. I sincerely hope that these collaborationswill continue in the future.

Moreover, I am especially indebted to all my friends and colleagues at theresearch groups at TU Vienna and ETH Zurich, which were both from a scien-tific and personal point of view perfect environments for my research. Seeingtruly good friends every day made the hard periods during my PhD studies a

v

vi Acknowledgment

lot easier. My thanks go to my roommates in Vienna, Sara Karlsson, Anto-nis Papapantoleon, Takahiro Tsuchiya and to my colleagues in Zurich, OzanAkdogan, David Belius, Matteo Casserini, Christoph Czichowsky, PhilippDorsek, Nicoletta Gabrielli, Selim Gokay, Georg Grafendorfer, Martin Herde-gen, Blanka Horvath, Florian Leisch, Marcel Nutz, Anja Richter, Stefan Tappeand Dejan Veluscek. Thank you all for the coffees, beers, laughs, mathemati-cal discussions and long evening sessions.

Finally, I gratefully acknowledge the financial support by ETH Zurich andthe START price project at TU Vienna.

In concluding, I would like to thank my friends and my family, in particularRenaud, who has always patiently listened to my problems of mathematicaland non-mathematical nature and whose loving company, support and encour-agement over the last years considerably helped me to accomplish the task ofwriting this thesis.

Contents

Abstract i

Kurzfassung iii

Acknowledgment v

Introduction 1

I Affine Processes 7

1 Affine Processes on General State Spaces 91.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2 Cadlag Version . . . . . . . . . . . . . . . . . . . . . . . . . . 141.3 Right-Continuity of the Filtration . . . . . . . . . . . . . . . . 281.4 Semimartingale Property . . . . . . . . . . . . . . . . . . . . . 301.5 Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2 Affine Processes on Proper Convex Cones 452.1 Definition of Cone-valued Affine Processes . . . . . . . . . . . 452.2 Feller Property and Regularity . . . . . . . . . . . . . . . . . . 492.3 Necessary Conditions . . . . . . . . . . . . . . . . . . . . . . . 52

2.3.1 Levy Khintchine Form of F and R . . . . . . . . . . . 532.3.2 Parameter Restrictions . . . . . . . . . . . . . . . . . . 572.3.3 Quasi-monotonicity . . . . . . . . . . . . . . . . . . . . 58

2.4 The Generalized Riccati Equations . . . . . . . . . . . . . . . 602.5 Construction of Affine Pure Jump Processes . . . . . . . . . . 63

3 Affine Processes on Symmetric Cones 673.1 Symmetric Cones and Euclidean Jordan Algebras . . . . . . . 673.2 Refinement of the Necessary Conditions . . . . . . . . . . . . . 70

3.2.1 Representation of the Diffusion Part . . . . . . . . . . 70

vii

viii Contents

3.2.2 Linear Jump Coefficient . . . . . . . . . . . . . . . . . 72

3.2.3 The Special Role of the Constant Drift Part . . . . . . 75

3.3 Discussion of the Admissibility Conditions . . . . . . . . . . . 82

3.3.1 Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . 83

3.3.2 Drift . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

3.3.3 Killing . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.3.4 Jumps . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.4 Construction of Affine Processes on Symmetric Cones . . . . . 86

3.4.1 Construction of Affine Diffusion Processes . . . . . . . 86

3.4.2 Existence of Affine processes on Symmetric cones . . . 98

3.5 Wishart Distribution . . . . . . . . . . . . . . . . . . . . . . . 101

3.5.1 Central Wishart Distribution . . . . . . . . . . . . . . 101

3.5.2 Non-central Wishart distribution . . . . . . . . . . . . 103

3.6 Relation to Infinitely Divisible Distributions . . . . . . . . . . 106

3.7 Results for Positive Semidefinite Matrices . . . . . . . . . . . . 110

II Polynomial Processes 113

4 Characterization and Relation to Semimartingales 115

4.1 Definition and Characterization . . . . . . . . . . . . . . . . . 116

4.2 Polynomial Semimartingales . . . . . . . . . . . . . . . . . . . 123

4.3 Characterization by means of the Extended Generator . . . . 132

4.3.1 Proof of Theorem 4.1.8 (iv) ⇒ (iii) . . . . . . . . . . . 132

4.3.2 Semimartingales which are Polynomial Processes . . . 133

4.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

III Applications 141

5 Multivariate Affine Volatility Models 143

5.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

5.2 Form of F and R . . . . . . . . . . . . . . . . . . . . . . . . . 147

5.3 Conservativeness and Martingale Property . . . . . . . . . . . 153

5.4 Semimartingale Representation of Affine Volatility Models . . 156

5.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

5.5.1 Multivariate Heston Model . . . . . . . . . . . . . . . . 163

5.5.2 Multivariate Barndorff-Nielsen-Shepard Model . . . . . 164

Contents ix

6 Applications of Polynomial Processes 1676.1 Moment Calculation . . . . . . . . . . . . . . . . . . . . . . . 167

6.1.1 Generalized Method of Moments . . . . . . . . . . . . 1686.2 Pricing - Variance Reduction . . . . . . . . . . . . . . . . . . . 169

Appendix 173

A Symmetric Cones and Euclidean Jordan Algebras 173A.1 Important Definitions . . . . . . . . . . . . . . . . . . . . . . . 173

A.1.1 Determinant, Trace and Inverse . . . . . . . . . . . . . 173A.1.2 Idempotents, Spectral and Peirce Decomposition . . . . 174A.1.3 Classification of Simple Euclidean Jordan Algebras . . 177

A.2 Some Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

Bibliography 181

Introduction

The aim of this thesis is to draw a comprehensive picture of the theory ofaffine and polynomial processes. Both are particular classes of continuous-timeMarkov processes, whose specific properties are determined by the followingassumptions on the associated semigroups: in the case of affine processes, theFourier-Laplace transform of the marginal distributions is supposed to dependin an exponential affine way on the initial state; in the case of polynomialprocesses, the semigroup is assumed to map polynomials into polynomials,that is, the expected value of any polynomial is again a polynomial in theinitial state. This implies in particular that under moment assumptions, affineprocesses are a subclass of polynomial processes.

The common property of both kinds of processes is the high degree of ana-lytical tractability implied by the fact that the expected value of a large classof payoff functions is explicitly known. This is one reason why many modelsused in mathematical finance fall under the setting of affine or polynomialprocesses. Moreover, as a particular class of jump diffusions, they allow for si-multaneous modeling of diffusive, jump and stochastic volatility phenomena ina multivariate setting, while remaining analytically tractable in a remarkableway.

From a mathematical point of view the affine class contains Levy processes,Ornstein-Uhlenbeck processes, continuous branching processes and their mul-tidimensional generalizations such as matrix-valued Wishart processes. Be-yond that, processes with quadratic diffusion coefficients, such as the Ja-cobi process or Levy-driven stochastic differential equations with affine vectorfields, are covered by the setting of polynomial processes.

The analysis of affine processes dates back to 1971, when Kawazu andWatanabe [1971] studied continuous-time limits of Galton-Watson branchingprocesses with immigration. These processes correspond exactly to the one-dimensional affine processes on the positive real line, which were taken up laterin the short-rate model of Cox, Ingersoll, and Ross [1985] and the stochasticvolatility model of Heston [1993]. The need for more complex models pro-

1

2 Introduction

gressively led to the introduction of higher dimensional affine jump diffusionson the so-called canonical state space, that is, Rm

+ × Rn−m, see, e.g., Dai andSingleton [2000], Duffie and Kan [1996], Duffie, Pan, and Singleton [2000].These affine jump diffusions are included in the setting of affine processeson Rm

+ × Rn−m, which are defined as continuous-time Markov processes withthe mentioned exponential affine property of the characteristic function. Acomplete characterization of these processes in terms of semimartingale char-acteristics and Markov generators was subsequently provided by Duffie et al.[2003].

The development of multivariate stochastic volatility models has recentlygiven rise to applications of affine processes on non-canonical state spaces, inparticular on the cone of positive semidefinite matrices. Such matrix-valuedaffine processes seem to have been studied systematically for the first timein the literature by Bru [1989, 1991], who introduced the so-called Wishartprocesses, which are multidimensional analogs of squared Bessel processes.

A natural generalization of positive semidefinite matrices are so-called sym-metric cones, which are cones of squares in Jordan algebras (see the workof Faraut and Koranyi [1994]). Affine diffusion processes thereon were intro-duced by Grasselli and Tebaldi [2008] with the objective to give some generalresults on the solvability of the corresponding Riccati differential equations.The setting of symmetric cones is taken up in this thesis.

State spaces whose boundary is described by a quadratic polynomial havebeen considered by Spreij and Veerman [2010]. It turns out that up to iso-morphisms the only possible state spaces of this form are parabolic ones (seealso Duffie et al. [2003, Section 12]) and the symmetric Lorentz cone. Let usremark that the boundary of other symmetric cones is in general described bypolynomials of higher degree.

Apart from these recent developments, affine processes have mainly beenstudied on the particular state space Rm

+ × Rn−m. Building on the resultsof affine processes on the canonical state space, the purpose of this thesis istwofold: first, the analysis of affine processes on general state spaces, in par-ticular convex cones and the subclass of symmetric cones; second, the studyof analytically tractable extensions of the affine class, which leads to the in-troduction of polynomial processes.

As mentioned before, the first line of research is motivated by multivariateaffine stochastic volatility models, which consist of a d-dimensional logarithmicprice process Y and an instantaneous stochastic covariation process X. Thesemodels correspond to particular affine processes on the mixed state spaceS+d × Rd, where S+

d denotes the cone of positive semidefinite d× d matrices.The first crucial property needed to study such models is the differentiabil-

3

ity of the Fourier-Laplace transform of the joint process (X, Y ) with respect totime, which is called regularity. This implies in particular that the (extended)cumulant generating function of (X, Y ) is given as solution of a system of or-dinary differential equations, upon which the analytical tractability of affinestochastic volatility models is based. The main theorem of Chapter 1 estab-lishes regularity for all affine processes on any state space. An alternative proofof this result was obtained by Keller-Ressel, Schachermayer, and Teichmann[2011]. The same authors also proved regularity on the canonical state space(see Keller-Ressel, Schachermayer, and Teichmann [2010]) by using differentmethods which exploit the structure of Rm

+ × Rn−m and cannot be extendedto general state spaces.

On the way to this result, we show additionally that every affine processadmits a cadlag version and is a semimartingale up to its lifetime, whose(absolutely continuous) characteristics depend in an affine way on the statevariables. This implies in particular that the derivative of the Fourier-Laplacetransform at t = 0 is of Levy-Khintchine form. The results of this chapterwill be published in Cuchiero and Teichmann [2011].

In order to further characterize affine processes, in particular in view of thequestion of existence, we assume the state space to be a closed convex cone inChapter 2. This allows us to prove the Feller property and existence of affinepure jump processes without diffusion component. The latter result relies onthe global existence and uniqueness of the corresponding generalized Riccatidifferential equations, by means of which the Fourier-Laplace transform isdetermined. Indeed, in the case of affine pure jump processes the solutions ofthese equations can be recognized as cumulant generating functions of sub-stochastic measures, as done in Duffie et al. [2003, Section 7]. This then settlesthe existence question.

For the characterization of general affine processes with diffusion compo-nent we further strengthen the assumptions on the state space by assumingthe setting of symmetric cones. This framework contains the cone of positivesemidefinite matrices, which is – in view of multivariate stochastic volatilitymodeling – particularly relevant for practical applications. Other examplesare Rm

+ , the Hermitian matrices or the Lorentz cone. This assumption on thestate space allows us to provide a full characterization of affine processes on ir-reducible symmetric cones. Our results of Chapter 3 show that the parametersof the semimartingale characteristics satisfy some well-determined admissibil-ity conditions, which differ in particular with regard to the constant drift partfrom those on the canonical state space. Conversely, and more importantlyfor applications, we show that for any admissible parameter set there exists aunique well-behaved affine process.

In the case of positive semidefinite matrices, where affine processes have

4 Introduction

already been studied in the context of stochastic volatility models, our findingsextend the model class since a more general drift and jumps are possible. Onthe other hand, we also establish the exact assumptions under which affineprocesses on S+

d actually exist.The existence proof is based on the result that the solutions of the general-

ized Riccati differential equations of a pure diffusion process with a particulardrift can be recognized as cumulant generating functions of the non-centralWishart distribution. This distribution is well studied on the cone of positivesemidefinite matrices (see, e.g., Letac and Massam [2004]), but for generalsymmetric cones only a few results on the central Wishart distribution areavailable in the literature. For example, the form of the density functionhas not been provided so far and is derived in Section 3.5. Beyond that,we prove that any stochastically continuous infinitely decomposable Markovprocess on a symmetric cone, whose dimension is greater than 2, is affinewith zero diffusion, and vice versa. This is a consequence of the fact that theWishart distribution is not infinitely divisible. The findings of Chapter 2 and 3generalize the results on positive semidefinite matrices obtained in Cuchiero,Filipovic, Mayerhofer, and Teichmann [2011a] to the setting of (symmetric)cones and will be published in Cuchiero, Keller-Ressel, Mayerhofer, and Te-ichmann [2011b]. For reasons of practical relevance and potential applicationsin mathematical finance, we restate the main theorems in the specific contextof positive semidefinite matrices (see Section 3.7). This also summarizes theresults found in Cuchiero et al. [2011a], of which some parts are now proveddifferently. In particular, in Cuchiero et al. [2011a] the question of existenceof affine processes is handled by solving the associated martingale problem(see Cuchiero et al. [2011a, Proposition 5.9]), whereas it is here reduced viaTrotter’s product formula to the existence of pure diffusion and pure jumpprocesses.

The second line of research, namely a generalization of affine processes,is based on the observation that the expected value of a polynomial of anaffine process is again a polynomial in the initial value. Taking this propertyas definition of a class of Markov processes, which we call polynomial, givesrise to an analytically tractable extension of the affine class, containing forexample processes with quadratic diffusion coefficients. We can thus describea class of processes for which it is easy and efficient to compute moments of allorders, even though neither their probability distribution nor their character-istic function needs to be known. Indeed, the calculation of (mixed) momentsonly requires the computation of matrix exponentials, where the entries of thecorresponding matrix can be easily deduced from the (extended) infinitesimalgenerator. This property of polynomial processes has already been exploited

5

by Zhou [2003] for GMM estimation of one-dimensional jump-diffusions. InChapter 4 we formally introduce this class of polynomial processes, establisha relationship to semimartingales and give conditions on the (extended) in-finitesimal generator such that a Markov process is polynomial. This chapterwill be published in Cuchiero, Keller-Ressel, and Teichmann [2010].

The last part of the thesis is devoted to applications of affine and polyno-mial processes. In Chapter 5 we study multivariate affine stochastic volatilitymodels, where we establish necessary and sufficient conditions on the param-eters describing the semimartingale characteristics such that the (discounted)price processes are martingales. In particular, we prove the necessity of acertain correlation structure between the Brownian motions driving the loga-rithmic price processes and the covariation process. The results of this chapterwill be published in Cuchiero [2011]. Chapter 6 then deals with applicationsof polynomial processes, especially how the analytical knowledge of momentscan be exploited for option pricing. This chapter is part of Cuchiero et al.[2010].

Notation

For the stochastic background and notation we refer to standard text bookssuch as Jacod and Shiryaev [2003] and Revuz and Yor [1999].

We write R+ for [0,∞), R++ for (0,∞) and Q+ for nonnegative rationalnumbers. Moreover, V always denotes some finite dimensional real vectorspace with scalar product 〈·, ·〉. The symmetric matrices and the positivesemidefinite matrices over V are denoted by S(V ) and S+(V ), respectively,while L(V ) corresponds to the space of linear maps on V . In the case V = Rd,we write Sd and S+

d for S(V ) and S+(V ), respectively, and Id denotes the d×didentity matrix.

Throughout this thesis, we shall consider the following function spaces formeasurable U ⊆ V , whose Borel σ-algebra is denoted by B(U). We writeC(U) for the space of (complex-valued) continuous functions f on U , Cb(U)for the space of (complex-valued) bounded continuous functions on U , Cc(U)for the space of functions f ∈ C(U) with compact support and C0(U) forthe Banach space of functions f ∈ C(U) with lim‖x‖→∞ f(x) = 0 and norm‖f‖∞ = supx∈U |f(x)|.

Part I

Affine Processes

7

Chapter 1

Affine Processes on GeneralState Spaces

We define affine processes as a particular class of time-homogeneous Markovprocesses with state space D ⊆ V , some closed, non-empty subset of an n-dimensional real vector space V with scalar product 〈·, ·〉. To clarify notation,we find it useful to recall in the first section the basic ingredients of thetheory of time-homogeneous Markov processes and the particular conventionsbeing made in this thesis (compare Blumenthal and Getoor [1968, Chapter1.3], Chung and Walsh [2005, Chapter 1.2], Ethier and Kurtz [1986, Chapter4], Rogers and Williams [1994, Chapter 3, Definition 1.1]).

1.1 Definition

Throughout D denotes a closed subset of V and D its Borel σ-algebra. Sincewe shall not assume the process to be conservative, we adjoin to the statespace D a point ∆ /∈ D, called cemetery state, and set D∆ = D∪∆ as wellas D∆ = σ(D, ∆). We make the convention that ‖∆‖ := ∞, where ‖ · ‖denotes the norm induced by the scalar product 〈·, ·〉, and we set f(∆) = 0for any other function f on D.

Consider the following objects on a space Ω:

(i) a stochastic process X = (Xt)t≥0 taking values in D∆ such that

if Xs(ω) = ∆, then Xt(ω) = ∆ for all t ≥ s and all ω ∈ Ω; (1.1)

(ii) the filtration generated by X, that is, F0t = σ(Xs, s ≤ t), where we set

F0 =∨t∈R+F0t ;

(iii) a family of probability measures (Px)x∈D∆on (Ω,F0).

9

10 Chapter 1. Affine Processes on General State Spaces

Definition 1.1.1 (Markov process). A time-homogeneous Markov process

X =(Ω, (F0

t )t≥0, (Xt)t≥0, (pt)t≥0, (Px)x∈D∆

)with state space (D,D) (augmented by ∆) is a D∆-valued stochastic processsuch that, for all s, t ≥ 0, x ∈ D∆ and all bounded D∆-measurable functionsf : D∆ → R,

Ex[f(Xt+s)|F0

s

]= EXs [f(Xt)] , Px-a.s. (1.2)

Here, Ex denotes the expectation with respect to Px and (pt)t≥0 is a transitionfunction on (D∆,D∆). A transition function is a family of kernels pt : D∆ ×D∆ → [0, 1] such that

(i) for all t ≥ 0 and x ∈ D∆, pt(x, ·) is a measure on D∆ with pt(x,D) ≤ 1,pt(x, ∆) = 1− pt(x,D) and pt(∆, ∆) = 1;

(ii) for all x ∈ D∆, p0(x, ·) = δx(·), where δx(·) denotes the Dirac measure;

(iii) for all t ≥ 0 and Γ ∈ D∆, x 7→ pt(x,Γ) is D∆-measurable;

(iv) for all s, t ≥ 0, x ∈ D∆ and Γ ∈ D∆, the Chapman-Kolmogorov equationholds, that is,

pt+s(x,Γ) =

∫D∆

ps(x, dξ)pt(ξ,Γ).

If (Ft)t≥0 is a filtration with F0t ⊂ Ft, t ≥ 0, then X is a time-homogeneous

Markov process relative to (Ft) if (1.2) holds with F0s replaced by Fs.

We can alternatively think of the transition function as inducing a mea-surable contraction semigroup (Pt)t≥0 defined by

Ptf(x) := Ex[f(Xt)] =

∫D

f(ξ)pt(x, dξ), x ∈ D∆,

for all bounded D∆-measurable functions f : D∆ → R.

Remark 1.1.2. (i) Note that, in contrast to Duffie et al. [2003], we do notassume Ω to be the canonical space of all functions ω : R+ → D∆, butwork on some general probability space.

(ii) Since we have pt(x,Γ) = Px[Xt ∈ Γ] for all t ≥ 0, x ∈ D∆ and Γ ∈ D∆,property (ii) and (iii) of the transition function, imply Px[X0 = x] = 1for all x ∈ D∆ and measurability of the map x 7→ Px[Xt ∈ Γ] for allt ≥ 0 and Γ ∈ D∆.

1.1. Definition 11

For the following definition of affine processes, let us introduce the set Udefined by

U =u ∈ V + iV

∣∣ e〈u,x〉 is a bounded function on D. (1.3)

Clearly iV ⊆ U . Here, the set iV stands for purely imaginary elements and〈·, ·〉 is the extension of the real scalar product to V + iV without complexconjugation. Moreover, we denote by p the dimension of ReU and write〈ReU〉 for its (real) linear hull and 〈ReU〉⊥ for its orthogonal complementin V . The set i 〈ReU〉⊥ ⊂ U are then the purely imaginary direction of U .Finally, for some linear subspace W ⊂ V , ΠW : V → V denotes the orthogonalprojection on W , which is extended to V +iV by linearity, i.e., ΠW (v1+i v2) :=ΠWv1 + i ΠWv2.

Assumption 1.1.3. Recall that dimV = n. We require that the state spaceD contains at least n + 1 affinely independent elements x1, . . . , xn+1, that is,the n vectors (x1 − xj, . . . , xj−1 − xj, xj+1 − xj, . . . , xn+1 − xj) are linearlyindependent for every j ∈ 1, . . . , n+ 1.

We are now prepared to give our main definition:

Definition 1.1.4 (Affine process). A time-homogeneous Markov process Xrelative to some filtration (Ft) and with state space (D,D) (augmented by ∆)is called affine if

(i) it is stochastically continuous, that is, lims→t ps(x, ·) = pt(x, ·) weakly onD for every t ≥ 0 and x ∈ D, and

(ii) its Fourier-Laplace transform has exponential-affine dependence on theinitial state. This means that there exist functions Φ : R+×U → C andΨ : R+ × U → V + iV such that

Ex[e〈u,Xt〉

]= Pte

〈u,x〉 =

∫D

e〈u,ξ〉pt(x, dξ) = Φ(t, u)e〈Ψ(t,u),x〉, (1.4)

for all x ∈ D and (t, u) ∈ R+ × U .

Remark 1.1.5. (i) The above definition differs in three crucial details fromthe definitions given in Duffie et al. [2003, Definition 2.1, Definition12.1].1

First, therein the right hand side of (1.4) is defined in terms of a functionφ(t, u), namely as eφ(t,u)+〈Ψ(t,u),x〉, such that the function Φ(t, u) in our

1In Definition 2.1 affine processes on the canonical state space D = Rm+ × Rn−m are

considered, whereas in Definition 12.1 the state space D can be an arbitrary subset of Rn.


definition corresponds to eφ(t,u).2 Our definition is in line with the onegiven in Kawazu and Watanabe [1971] and Keller-Ressel et al. [2010]and differs from the one in Duffie et al. [2003], as we do not requireΦ(t, u) 6= 0 a priori. However, since all affine processes on D = Rm

+ ×Rn−m are infinitely divisible (see Duffie et al. [2003, Theorem 2.15]),it turns out with hindsight that setting Φ(t, u) = eφ(t,u) is actually norestriction.

Second, we assume that the affine property holds for all u ∈ U , whereason the canonical state space D = Rm

+ × Rn−m it is restricted to iRn

(see Duffie et al. [2003]). This however turns out to imply the affineproperty also on U .

Third, in contrast to Duffie et al. [2003], we take stochastic continuity aspart of the definition of an affine process. We remark that there are sim-ple examples of Markov processes which satisfy Definition 1.1.4 (ii), butare not stochastically continuous (see Duffie et al. [2003, Remark 2.11]).

(ii) Note that Assumption 1.1.3 is no restriction, since we can always passto a lower dimensional ambient vector space if D does not contain n+ 1affinely independent elements.

(iii) Let us remark that in Section 1.2 we consider affine processes on thefiltered space (Ω,F0,F0

t ), where F0t denotes the natural filtration and

F0 =∨t∈R+F0t , as introduced above. However, we shall progressively

enlarge the filtration by augmenting with the respective null-sets.

Before deducing the first properties of Φ and Ψ from the above definition,let us introduce the sets

Um =

u ∈ V + iV | sup

x∈De〈Reu,x〉 ≤ m

, m ≥ 1,

and note that U =⋃m≥1 Um and iV ⊆ Um for all m ≥ 1.

Proposition 1.1.6. Let X be an affine process relative to some filtration (Ft).Then the functions Φ and Ψ have the following properties:

(i) For every m ≥ 1, Φ and Ψ can be chosen to be jointly continuouson Qm = (t, u) ∈ R+ × Um |Φ(s, u) 6= 0, for all s ∈ [0, t]. Thisthen yields a unique specification of Φ and Ψ on Q = (t, u) ∈ R+ ×U |Φ(s, u) 6= 0, for all s ∈ [0, t].

2Note that the function Ψ in our definition is denoted by ψ in Duffie et al. [2003,Definition 2.1].

1.1. Definition 13

(ii) Ψ maps the set O = (t, u) ∈ R+ × U |Φ(t, u) 6= 0 to U .

(iii) Φ(0, u) = 1 and Ψ(0, u) = u for all u ∈ U .

(iv) The functions Φ and Ψ satisfy the semiflow property: Let u ∈ U andt, s ≥ 0. Suppose that Φ(t + s, u) 6= 0, then also Φ(t, u) 6= 0 andΦ(s,Ψ(t, u)) 6= 0 and we have

Φ(t+ s, u) = Φ(t, u)Φ(s,Ψ(t, u)),

Ψ(t+ s, u) = Ψ(s,Ψ(t, u)).(1.5)

Proof. It follows e.g. from Bauer [1996, Lemma 23.7] that stochastic continuityof X implies joint continuity of (t, u) 7→ Pte

〈u,x〉 on R+ × Um for all x ∈ D.Hence (t, u) 7→ Φ(t, u)e〈Ψ(t,u),x〉 is jointly continuous on R+ × Um for everyx ∈ D. By Assumption 1.1.3 on the state space D, this in turn yields aunique continuous choice of the functions (t, u) 7→ Φ(t, u) and (t, u) 7→ Ψ(t, u)on Qm. (compare Keller-Ressel et al. [2011, Proposition 2.4] for details).

Concerning (ii), let (t, u) ∈ O = (t, u) ∈ R+ × U |Φ(t, u) 6= 0. Since∣∣Φ(t, u)e〈Ψ(t,u),x〉∣∣ =∣∣Ex [e〈u,Xt〉]∣∣ ≤ Ex

[∣∣e〈u,Xt〉∣∣]is bounded on D and as Φ(t, u) 6= 0, we conclude that Ψ(t, u) ∈ U .

Assertion (iii) follows simply from

e〈u,x〉 = Ex[e〈u,X0〉

]= Φ(0, u)e〈Ψ(0,u),x〉.

Assumption Φ(t+ s, u) 6= 0 in (iv) implies

Ex[e〈u,Xt+s〉

]= Φ(t+ s, u)e〈Ψ(t+s,u),x〉 6= 0. (1.6)

By the law of iterated expectations and the Markov property, we thus have

Ex[e〈u,Xt+s〉

]= Ex

[Ex[e〈u,Xt+s〉

∣∣∣Fs]] = Ex[EXs

[e〈u,Xt〉

]]. (1.7)

If Φ(t, u) = 0 or Φ(s,Ψ(t, u)) = 0, then the inner or the outer expectationevaluates to 0. This implies that the whole expression is 0, which contra-dicts (1.6). Hence Φ(t, u) 6= 0 and Φ(s,Ψ(t, u)) 6= 0 and we can write (1.7)as

Ex[e〈u,Xt+s〉

]= Ex

[Φ(t, u)e〈Ψ(t,u),Xs〉

]= Φ(t, u)Φ(s,Ψ(t, u))e〈Ψ(s,Ψ(t,u)),x〉.

Comparing with (1.6) yields the claim.

Remark 1.1.7. Henceforth, the symbols Φ and Ψ always correspond to theunique continuous choice established in Proposition 1.1.6.


1.2 Cadlag Version

The aim of this section is to show that the definition of an affine process alreadyimplies the existence of a cadlag version. Indeed, for every fixed x ∈ D, wefirst establish that for Px-almost every ω

t 7→MT,ut (ω) := Φ(T − t, u)e〈Ψ(T−t,u),Xt(ω)〉, t ∈ [0, T ],

is the restriction to Q+ ∩ [0, T ] of a cadlag function for almost all (T, u) ∈(0,∞)×U , in the sense that MT,u

t = 0 if Φ(T−t, u) = 0. This is an applicationof Doob’s regularity theorem for supermartingales, where we can conclude –using Fubini’s theorem – that there exists a Px-null-set outside of which weobserve appropriately regular trajectories for almost all (T, u).

Proposition 1.2.1. Let x ∈ D be fixed and let X be an affine process relativeto (F0

t ). Then

limq∈Q+q↓t

MT,uq = lim

q∈Q+q↓t

Φ(T − q, u)e〈Ψ(T−q,u),Xq〉, t ∈ [0, T ],

exists Px-a.s. for almost all (T, u) ∈ (0,∞)×U and defines a cadlag functionin t.

Proof. In order to prove this result, we adapt parts of the proof of Protter[2005, Theorem I.4.30] to our setting. Due to the law of iterated expectations

MT,ut = Φ(T − t, u)e〈Ψ(T−t,u),Xt〉 = Ex

[e〈u,XT 〉

∣∣F0t

], t ∈ [0, T ],

is a (complex-valued) (F0t ,Px)-martingale for every u ∈ U and every T >

0. From Doob’s regularity theorem (see, e.g., Rogers and Williams [1994,Theorem II.65.1]) it then follows that, for any fixed (T, u), the function t 7→MT,u

t (ω), with t ∈ Q+ ∩ [0, T ], is the restriction to Q+ ∩ [0, T ] of a cadlagfunction for Px-almost every ω. Define now the set

Γ = (ω, T, u) ∈ Ω× (0,∞)× U | t 7→MT,ut (ω), t ∈ Q+ ∩ [0, T ],

is not the restriction of a cadlag function. (1.8)

Then Γ is a F0 ⊗ B((0,∞)× U)-measurable set. Due to the above argumentconcerning regular versions of (super-)martingales,

∫Ω

1Γ(ω, T, u)Px(dω) = 0for any (T, u) ∈ (0,∞)× U . By Fubini’s theorem, we therefore have∫

Ω

∫(0,∞)×U

1Γ(ω, T, u)dλPx(dω) =

∫(0,∞)×U

∫Ω

1Γ(ω, T, u)Px(dω) dλ = 0,

where λ denotes the Lebesgue measure. Hence, for Px-almost every ω, t 7→MT,u

t (ω), t ∈ Q+ ∩ [0, T ], is the restriction of a cadlag function for λ-almostall (T, u) ∈ (0,∞)× U , which proves the result.

1.2. Cadlag Version 15

Having established path regularity of the martingales MT,u, we want todeduce the same for the affine process X. This is the purpose of the subsequentlemmas and propositions, for which we need to introduce the following setsΩ ⊆ Ω, T ⊆ (0,∞) and V ⊆ iV :

Ω is the projection of Ω× (0,∞)× iV \ Γ onto Ω, (1.9)

T is the projection of Ω× (0,∞)× iV \ Γ onto (0,∞), (1.10)

V is the projection of Ω× (0,∞)× U \ Γ onto U , (1.11)

where Γ is given in (1.8). Denoting by Fx the completion of F0 with respect to

Px, let us remark that the measurable projection theorem implies that Ω ∈ Fxand by the above proposition we have Px[Ω] = 1. Finally, for some r > 0,we denote by K the intersection of V with the closed ball with center 0 andradius r, that is,

K := B(0, r) ∩ V :=u ∈ U | ‖Reu‖2 + ‖Imu‖2 ≤ r2

∩ V . (1.12)

Lemma 1.2.2. Consider the set K defined in (1.12) and the function Ψ givenin (1.4) with the properties of Proposition 1.1.6. Denote by p the dimensionof ReU . Let (u1, . . . , up) be linearly independent vectors in K ∩ ReU and let(up+1, . . . , un) be linearly independent vectors in Π〈ReU〉⊥K. Then there exists

some δ > 0 such that for every t ∈ [0, δ)

(Ψ(t, u1), . . . ,Ψ(t, up))

and(Π〈ReU〉⊥Ψ(t, up+1), . . . ,Π〈ReU〉⊥Ψ(t, un))

are linearly independent.

Proof. This is simply a consequence of the fact that Ψ(0, u) = u for all u ∈U ⊃ K and the continuity of t 7→ Ψ(t, u).

The following lemma is needed to prove Proposition 1.2.4 below which isessential for establishing the existence of a cadlag version of X.

Lemma 1.2.3. Let Ψ be given by (1.4) and assume that there exists someD-valued sequence (xk)k∈N such that

limk→∞

Π〈ReU〉xk =: limk→∞

yk (1.13)

exists finitely valued and

lim supk→∞

‖Π〈ReU〉⊥xk‖ =∞. (1.14)


Then we can choose a subsequence of (xk) denoted again by (xk): along thissequence there exist a finite number of mutually orthogonal directions gi ∈〈ReU〉⊥ of length 1 such that

xk −∑i

〈xk, gi〉gi

converges as k →∞ and 〈xk, gi〉 diverges as k →∞, where the rates of diver-gence are decreasing in i (see the proof for the precise statement). Further-more, there exist continuous functions R : R+ → R++ and λi : R+ → 〈ReU〉⊥such that

〈Ψ(t, u), gi〉 = 〈λi(t), u〉for all u ∈ Π〈ReU〉⊥ U with ‖Imu‖ < R(t).

Proof. Concerning the first assertion, we define – by choosing appropriatesubsequences, still denoted by (xk), – the directions of divergence in 〈ReU〉⊥inductively by

gr = limk→∞

xk −∑r−1

i=1 〈xk, gi〉gi‖xk −

∑r−1i=1 〈xk, gi〉gi‖

, (1.15)

as long as lim supk→∞ ‖xk −∑r−1

i=1 〈xk, gi〉gi‖ =∞. Notice that we can choosethe directions gi mutually orthogonal and the rates of divergence of 〈gi, xk〉decreasing in i.

For the second part of the statement, we adapt the proof of Keller-Resselet al. [2010, Lemma 3.1] to our situation, using in particular the existenceof a sequence in D with the properties (1.13) and (1.14). As characteristicfunction, the map iV 3 u 7→ Ex[e〈u,Xt〉] is positive definite for any x ∈ D andt ≥ 0. Define now for every u ∈ Π〈ReU〉⊥U ⊆ iV , x ∈ D and t ≥ 0 the function

Θ(u, t, x) =Ex[e〈u,Xt〉

]Φ(t, 0)e〈Π〈ReU〉Ψ(t,0),Π〈ReU〉x〉

=Φ(t, u)e〈Ψ(t,u),x〉

Φ(t, 0)e〈Π〈ReU〉Ψ(t,0),Π〈ReU〉x〉. (1.16)

As Ex[e〈0,Xt〉

]= Φ(t, 0)e〈Ψ(t,0),x〉 is real-valued and positive for all t ≥ 0,

we conclude – due to Assumption 1.1.3 and the continuity of the functionst 7→ Φ(t, 0) and t 7→ Ψ(t, 0) – that ImΦ(t, 0) = 0 and ImΨ(t, 0) = 0 for allt ≥ 0. In particular, the denominator in (1.16) is positive, which implies thatiV ⊇ Π〈ReU〉⊥U 3 u 7→ Θ(u, t, x) is a positive definite function for all t ≥ 0

and x ∈ D. Moreover, since Π〈ReU〉⊥Ψ(t, 0) is purely imaginary and thus inparticular 0 for all t ≥ 0, it follows that

Θ(0, t, x) = exp(〈Π〈ReU〉⊥Ψ(t, 0),Π〈ReU〉⊥x〉

)= 1


for all t ≥ 0 and x ∈ D. This together with the positive definiteness ofu 7→ Θ(u, t, x) yields

|Θ(u+ v, t, x)−Θ(u, t, x)Θ(v, t, x)|2 ≤ 1, u, v ∈ Π〈ReU〉⊥U , t ≥ 0, x ∈ D,(1.17)

(compare, e.g., Keller-Ressel et al. [2010, Lemma 3.2]). Let us now definey := Π〈ReU〉x and

Z1(u, v, y, t) =Φ(t, u+ v)e〈Π〈ReU〉Ψ(t,u+v),y〉

Φ(t, 0)e〈Π〈ReU〉Ψ(t,0),y〉 ,

Z2(u, v, y, t) =Φ(t, u)Φ(t, v)e〈Π〈ReU〉(Ψ(t,u)+Ψ(t,v)),y〉

Φ(t, 0)2e2〈Π〈ReU〉Ψ(t,0),y〉 ,

β1(u, v, t) = Im(Π〈ReU〉⊥Ψ(t, u+ v)),

β2(u, v, t) = Im(Π〈ReU〉⊥Ψ(t, u)) + Im(Π〈ReU〉⊥Ψ(t, v)),

r1(u, v, y, t) = |Z1| =∣∣∣∣Φ(t, u+ v)

Φ(t, 0)

∣∣∣∣ e〈Re(Π〈ReU〉(Ψ(t,u+v)−Ψ(t,0))),y〉,

r2(u, v, y, t) = |Z2| =∣∣∣∣Φ(t, u)Φ(t, v)

Φ(t, 0)2

∣∣∣∣ e〈Re(Π〈ReU〉(Ψ(t,u)+Ψ(t,v)−2Ψ(t,0))),y〉,

α1(u, v, y, t) = arg(Z1) = arg

(Φ(t, u+ v)

Φ(t, 0)

)+ 〈Im(Π〈ReU〉Ψ(t, u+ v)), y〉,

α2(u, v, y, t) = arg(Z2) = arg

(Φ(t, u)Φ(t, v)

Φ(t, 0)2

)+ 〈Im(Π〈ReU〉(Ψ(t, u) + Ψ(t, v)), y〉.

Using (1.17) and the same arguments as in Keller-Ressel et al. [2010, Lemma3.1], we then obtain

1 ≥∣∣∣r1e

i(α1+〈β1,Π〈ReU〉⊥x〉) − r2ei(α2+〈β2,Π〈ReU〉⊥x〉)

∣∣∣2= r2

1 + r22 − 2r1r2 cos(α1 − α2 + 〈β1 − β2,Π〈ReU〉⊥x〉)

≥ 2r1r2(1− cos(α1 − α2 + 〈β1 − β2,Π〈ReU〉⊥x〉)),

whence

r1(u, v, y, t)r2(u, v, y, t)

×(1−cos(α1(u, v, y, t)−α2(u, v, y, t)+〈β1(u, v, t)−β2(u, v, t),Π〈ReU〉⊥x〉)) ≤1

2.

(1.18)


Define now

R(t, y) = sup

ρ ≥ 0 | r1(u, v, y, t)r2(u, v, y, t) >

3

4for u, v ∈ Π〈ReU〉⊥U

with ‖Imu‖ ≤ ρ and ‖Imv‖ ≤ ρ

.

Note that R(t, y) > 0 for all (t, y) ∈ R+ × Π〈ReU〉D, which follows from thefact that r1(0, 0, y, t) = r2(0, 0, y, t) = 1 and the continuity of the functions(u, v) 7→ r1(u, v, y, t)r2(u, v, y, t). Continuity of (t, y) 7→ r1(u, v, y, t)r2(u, v, y, t)also implies that (t, y) 7→ R(t, y) is continuous. Set now R(t) := infk R(t, yk)where yk = Π〈ReU〉xk. Then (1.13) implies that R(t) > 0 for all t ≥ 0. Letnow t be fixed and g1 given by (1.15). Suppose that

〈β1(u∗, v∗, t)− β2(u∗, v∗, t), g1〉 6= 0

for some u∗, v∗ ∈ Π〈ReU〉⊥U with ‖Imu∗‖ < R(t) and ‖Imv∗‖ < R(t). Thendue to the continuity of β1 and β2, there exists some δ > 0 such that for allu, v in a neighborhood Oδ of (u∗, v∗) defined by

Oδ =

(u, v) ∈ (Π〈ReU〉⊥U)2 | ‖Im(u− u∗)‖ < δ, ‖Im(v − v∗)‖ < δ and

‖Imu‖ < R(t), ‖Imv‖ < R(t),

we also have 〈β1(u, v, t) − β2(u, v, t), g1〉 6= 0. Moreover, there exist some(u, v) ∈ Oδ and some k ∈ N such that

cos(α1(u, v, yk, t)− α2(u, v, yk, t) + 〈β1(u, v, t)− β2(u, v, t),Π〈ReU〉⊥xk〉) ≤1

3,

(1.19)

since yk stays in a bounded set and Π〈ReU〉⊥xk explodes with highest divergence

rate in direction g1. However, inequality (1.19) now implies that

r1(u, v, yk, t)r2(u, v, yk, t)

×

(1− cos

(α1(u, v, yk, t)− α2(u, v, yk, t)

+ 〈β1(u, v, t)− β2(u, v, t),Π〈ReU〉⊥xk〉))

>1

2,

which contradicts (1.18). Since g1 corresponds to the direction of the highestdivergence rate, we thus conclude that

〈β1(u, v, t)− β2(u, v, t), g1〉 = Im(〈Ψ(t, u+ v)−Ψ(t, u)−Ψ(t, v), g1〉) = 0


for all u, v with ‖Imu‖ < R(t) and ‖Imv‖ < R(t). Continuity of u 7→ Ψ(t, u)therefore implies that u 7→ 〈Ψ(t, u), g1〉 is a linear function. Hence there existsa continuous curve of (real) vectors λ1(t) ∈ 〈ReU〉⊥ such that

〈Ψ(t, u), g1〉 = 〈λ1(t), u〉

for all u ∈ Π〈ReU〉⊥ U with ‖Imu‖ < R(t).We can now proceed inductively for the remaining directions of divergence

gi. Indeed, assume that 〈β1(u, v, t) − β2(u, v, t), gi〉 = 0 for all i ≤ r − 1 andall u, v with ‖Imu‖ < R(t) and ‖Imv‖ < R(t). Then reapting the above stepsallows us to conclude that 〈β1(u, v, t) − β2(u, v, t), gr〉 = 0 for all u, v with‖Imu‖ < R(t) and ‖Imv‖ < R(t) as well, which yields the assertion.

Consider now the set K defined in (1.12). Since (t, u) 7→ Φ(t, u)e〈Ψ(t,u),x〉

is jointly continuous on Qm for every m ≥ 1 with Φ(0, u) = 1 and Ψ(0, u) = u(see Proposition 1.1.6), it follows that there exists some η > 0 such that forall t ∈ [0, η]

infu∈K|Φ(t, u)| > c and sup

u∈K‖ReΨ(t, u)‖2 + ‖ImΨ(t, u)‖2 < C, (1.20)

with some constants c and C. By fixing these constants and some linearlyindependent vectors in K as described in Lemma 1.2.2, we define

ε := min(η, δ), (1.21)

where δ > 0 is given in Lemma 1.2.2. Moreover, let t ≥ 0 be fixed. Then wedenote by ITt,ε the set

ITt,ε := (t, t+ ε) ∩ T , (1.22)

where T is defined in (1.10).We are now prepared to prove the following proposition, which is the main

ingredient in the existence proof of a cadlag version of X (see the proof ofTheorem (1.2.7) below).

Proposition 1.2.4. Let K and ITt,ε be the sets defined in (1.12) and (1.22).Consider the function Ψ given in (1.4) with the properties of Proposition 1.1.6.Let t ≥ 0 be fixed and consider a sequence (qk)k∈N with values in Q+ ∩ [0, t]such that qk ↑ t. Moreover, let (xqk)k∈N be a sequence with values in D∆∪∞.Here, ∞ corresponds to a “point at infinity” and D∆ ∪ ∞ is the one-pointcompactification of D∆.3 Then the following assertions hold:

3If the state space D is compact, we do not adjoin ∞ and only consider a sequencewith values in D∆.


(i) If for all (T, u) ∈ ITt,ε ×K

limk→∞

NT,uqk

:= limk→∞

e〈Ψ(T−qk,u),xqk 〉 (1.23)

exists finitely valued and does not vanish, then also limk→∞ xqk existsfinitely valued.

(ii) If there exist some (T, u) ∈ ITt,ε ×K such that

limk→∞

NT,uqk

:= limk→∞

e〈Ψ(T−qk,u),xqk 〉 = 0,

then we have limk→∞ ‖xqk‖ =∞.

Moreover, let (qTk )k∈N,T∈ITt,ε be a family of sequences with values in Q+ ∩ [t, T ]

such that qTk ↓ t for every T ∈ ITt,ε and the additional property that for everyS, T ∈ ITt,ε, with S < T , there exists some index N ∈ N such that, for allk ≥ N , qSk−N = qTk . Then the above assertions hold true for these right limitswith qk replaced by qTk .

Remark 1.2.5. Concerning assertion (ii) of Proposition 1.2.4, note that,e.g. in the case qk ↑ t, limk→∞ ‖xqk‖ = ∞ corresponds either to explosion orto the possibility that there exists some index N ∈ N such that xqk = ∆ forall k ≥ N . In the latter case we also have, due to the convention ‖∆‖ = ∞,limk→∞ ‖xqk‖ =∞.

Proof. We start by proving the first assertion (i). Let T ∈ ITt,ε be fixed anddefine for all u ∈ K

A(u) := lim supk→∞

〈ReΨ(T − qk, u), xqk〉 , a(u) := lim infk→∞

〈ReΨ(T − qk, u), xqk〉 .

Then there exist subsequences (xqkm ) and (xqkl ) such that4

A(u) = limm→∞

⟨ReΨ(T − qkm , u), xqkm

⟩,

a(u) = liml→∞

⟨ReΨ(T − qkl , u), xqkl

⟩.

First note that A(u) and a(u) exist finitely valued for all u ∈ K. Indeed, ifthere is some u ∈ K such that A(u) = ±∞ or a(u) = ±∞, then the limit

4Note that these subsequences depend on u. For notational convenience we howeversuppress the dependence on u.


of NT,uqk

does not exist, or limk→∞NT,uqk

is either 0 or +∞, which contradictsassumption (1.23). We now define

r1(u) = limm→∞

exp(⟨

ReΨ(T − qkm , u), xqkm⟩),

r2(u) = liml→∞

exp(⟨

ReΨ(T − qkl , u), xqkl

⟩),

ϕm(u) =⟨ImΨ(T − qkm , u), xqkm

⟩,

ϕl(u) =⟨

ImΨ(T − qkl , u), xqkl

⟩.

Then the limits of cos(ϕm(u)), cos(ϕl(u)), sin(ϕm(u)) and sin(ϕl(u)) necessar-ily exist and

r1(u) limm→∞

cos(ϕm(u)) = r2(u) liml→∞

cos(ϕl(u)),

r1(u) limm→∞

sin(ϕm(u)) = r2(u) liml→∞

sin(ϕl(u)).

This yields r1(u) = r2(u) for all u ∈ K, since

limm→∞

(cos2(ϕm(u)) + sin2(ϕm(u)

)= lim

l→∞

(cos2(ϕl(u)) + sin2(ϕl(u)

)= 1.

In particular, we have proved that limk→∞ 〈ReΨ(T − qk, u), xqk〉 exists finitelyvalued for all (T, u) ∈ ITt,ε × K. By choosing linear independent vectors(u1, . . . , up) ∈ K ∩ ReU , it thus follows that

limk→∞

Π〈ReU〉xqk

exists finitely valued.Therefore it only remains to focus on Π〈ReU〉⊥xqk . From the above, we

know in particular that for all (T, u) ∈ ITt,ε ×K

limk→∞

e

⟨Π〈ReU〉⊥Ψ(T−qk,u),Π〈ReU〉⊥xqk

⟩(1.24)

exists finitely valued and does not vanish. This implies that for all (T, u) ∈ITt,ε ×K

Im⟨

Π〈ReU〉⊥Ψ(T − qk, u),Π〈ReU〉⊥xqk

⟩= αk(T, u) + 2πzk(T, u), (1.25)

where αk(T, u) ∈ [−π, π), α(T, u) := limk→∞ αk(T, u) exists finitely valuedand (zk(T, u))k∈N is a sequence with values in Z, which a priori does notnecessarily have a limit and/or limk→∞ zk(T, u) = ±∞.


Let us first assume that

lim supk→∞

‖Π〈ReU〉⊥xqk‖ =∞. (1.26)

Then we are exactly in the situation of Lemma 1.2.3 and the above limit (1.24)– after possibly selecting a subsequence such that xqk−

∑i gi 〈gi, xqk〉 converges

as k →∞ – can be written as

limk→∞

e

(∑i〈λi(T−qk),u〉〈gi,xqk 〉+

⟨Π〈ReU〉⊥Ψ(T−qk,u),xqk−

∑i gi 〈gi,xqk 〉

⟩)

for all u ∈ Π〈ReU〉⊥K with ‖Imu‖ < P (T ), where P (T ) is defined by P (T ) :=

infk R(T − qk) and R and the directions gi are given in Lemma 1.2.3. Notethat due to the strict positivity and continuity of R, P (T ) is strictly positiveas well. Furthermore, there exists some T ∗ ∈ ITt,ε and some set MT ∗ ⊆ u ∈Π〈ReU〉⊥K | ‖Imu‖ < P (T ∗),

∑i〈λi(T ∗ − t), u〉 6= 0 of positive finite measure

such that

limk→∞

∫M∗T

e

⟨Π〈ReU〉⊥Ψ(T ∗−qk,u),xqk−

∑i gi 〈gi,xqk 〉

⟩e(∑i〈λi(T ∗−qk),u〉〈gi,xqk 〉)du 6= 0.

(1.27)

However, it follows from the Riemann-Lebesgue Lemma that the previouslimit is zero, whence contradicting (1.27). We therefore conclude that

lim supk→∞

‖Π〈ReU〉⊥xqk‖ <∞.

This in turn implies that there exists some (T ∗, u∗) ∈ ITt,ε × K and N ∈ Nsuch that for all k ≥ N

Im⟨

Π〈ReU〉⊥Ψ(T ∗ − qk, u∗),Π〈ReU〉⊥xqk

⟩∈ (−π, π).

Indeed, this follows from the fact that for every u ∈ K and η > 0 there existssome T ∗ ∈ ITt,ε and N ∈ N such that for all k ≥ N

‖Im(Π〈ReU〉⊥Ψ(T ∗ − qk, u)− Π〈ReU〉⊥u)‖ ≤ η. (1.28)

For u∗ with ‖Im(Π〈ReU〉⊥u∗)‖ sufficiently small and k sufficiently large, we thus

have∣∣∣⟨Π〈ReU〉⊥Ψ(T ∗ − qk, u∗),Π〈ReU〉⊥xqk

⟩∣∣∣≤ (‖Im(Π〈ReU〉⊥u

∗)‖+ ‖Im(Π〈ReU〉⊥Ψ(T ∗ − qk, u∗)− Π〈ReU〉⊥u∗)‖)

× (lim supk→∞

‖Π〈ReU〉⊥xqk‖+ 1)

< π.


Hence

limk→∞

Im⟨

Π〈ReU〉⊥Ψ(T ∗ − qk, u∗),Π〈ReU〉⊥xqk

⟩= α(T ∗, u∗). (1.29)

As we can find n− p linear independent vectors up+1, . . . , un such that (1.29)is satisfied, we conclude using Lemma 1.2.2 that

limk→∞

Π〈ReU〉⊥xqk

exists finitely valued. This proves assertion (i).Concerning the second statement, observe that we have

limk→∞

e〈Ψ(T−qk,u),xqk 〉 = 0, (1.30)

if either explosion occurs or if xqN jumps to ∆ for some N ∈ N and xqk = ∆ forall k ≥ N . (This happens when the corresponding process is killed.) Indeed,since (1.30) is equivalent to limk→∞ e

〈ReΨ(T−qk,u),xqk 〉 = 0 and as ‖Ψ(T − t, u)‖is bounded on K due to the definition of ITt,ε, we necessarily have

limk→∞‖xqk‖ =∞.

In the case of a jump to ∆, this is implied by the conventions ‖∆‖ =∞ andf(∆) = 0 for any other function.

Similar arguments yield the assertion concerning right limits.

Using Proposition 1.2.1 and Proposition 1.2.4 above, we are now preparedto prove Theorem 1.2.7 below, which asserts the existence of a cadlag versionof X. Before stating this result, let us recall the notion of the (usual) aug-mentation of (F0

t ) with respect to Px, which guarantees the cadlag version tobe adapted.

Definition 1.2.6 (Usual augmentation). We denote by Fx the completionof F0 with respect to Px. A sub-σ-algebra G ⊂ Fx is called augmented withrespect to Px if G contains all Px-null-sets in Fx. The augmentation of F0

t withrespect to Px is denoted by Fxt , that is, Fxt = σ(F0

t ,N (Fx)), where N (Fx)denotes all Px-null-sets in Fx.

Theorem 1.2.7. Let X be an affine process relative to (F0t ). Then there

exists a process X such that, for each x ∈ D∆, X is a Px-version of X, whichis cadlag in D∆ ∪ ∞ (in D∆ respectively if D is compact) and an affineprocess relative to (Fxt ). As before, ∞ corresponds to a “point at infinity” andD∆ ∪ ∞ is the one-point compactification of D∆, if D is not compact.


Remark 1.2.8. We here establish the existence of a cadlag version X whosesample paths may take ∞ as left limiting value if D is not compact. A priori,we cannot identify Xs−(ω) with ∆, whenever ‖Xs−(ω)‖ = ∞. Indeed, Xt(ω)might become finitely valued for some t ≥ s. This issue is clarified in Theo-rem 1.2.10 below, where we prove that Px-a.s. ‖Xt‖ =∞ for all t ≥ s and all

s > 0 if ‖Xs−‖ =∞. In particular, this allows us to identify ∞ with ∆.

In the case Xs = ∆, which happens when the process is killed, Assump-tion (1.1) guarantees that Xt = ∆ for all t ≥ s and all s > 0.

Proof. It follows from Proposition 1.2.1 that for every ω ∈ Ω5, where Px[Ω] =1,

t 7→MT,ut (ω) := Φ(T − t, u)e〈Ψ(T−t,u),Xt(ω)〉, t ∈ [0, T ],

is the restriction to Q+ ∩ [0, T ] of a cadlag function for all (T, u) ∈ T × V .

Here, Ω, T and V are defined in (1.9), (1.10) and (1.11). Hence, for every

ω ∈ Ω and all (T, u) ∈ T × V , the limits

limq∈Q+q↑t

MT,uq (ω), lim

q∈Q+q↓t

MT,uq (ω) (1.31)

exist finitely valued for all t ∈ [0, T ].Let us now show that the same holds true for X. For notational con-

venience we first focus on left limits. Consider the sets K and ITt,ε definedin (1.12) and (1.22) and let t ≥ 0 be fixed. Take some sequence (qk)k∈N, asspecified in Proposition 1.2.4, such that qk ↑ t. Then there exists some N ∈ Nsuch that, for all k ≥ N and (T, u) ∈ ITt,ε × K, Φ(T − qk, u) 6= 0. This is aconsequence of the definition of ε (see (1.21)). Thus we can divide MT,u

qk(ω)

by Φ(T − qk, u) for all k ≥ N and (T, u) ∈ ITt,ε × K. By the continuity of

t 7→ Φ(t, u) and (1.31), it follows that, for every ω ∈ Ω, the limit

limk→∞

NT,uqk

(ω) := limk→∞

e〈Ψ(T−qk,u),Xqk (ω)〉

exists finitely valued for all (T, u) ∈ ITt,ε ×K. From Proposition 1.2.4 we thus

deduce that, for every ω ∈ Ω, the limit

limk→∞

Xqk(ω)

exists either finitely valued or limk→∞ ‖Xqk(ω)‖ = ∞. Using similar argu-ments yields the same assertion for right limits. Hence we can conclude thatPx-a.s.

Xt = limq∈Q+q↓t

Xq (1.32)

5Note that due to the measurable projection theorem, Ω ∈ Fx.


exists for all t ≥ 0 and defines a cadlag function in t.Let now Ω0 be the set of ω ∈ Ω for which the limit Xt(ω) exists for every

t and defines a cadlag function in t. Then, as a consequence of Rogers andWilliams [1994, Theorem II.62.7, Corollary II.62.12], Ω0 ∈ F0 and Px[Ω0] = 1

for all x ∈ D∆. For ω ∈ Ω\Ω0, we set Xt(ω) = ∆ for all t. Then X is a cadlag

process and Xt is F0-measurable for every t ≥ 0. Since X is assumed to bestochastically continuous, we have Xs → Xt in probability as s → t. Usingthe fact that convergence in probability implies almost sure convergence alonga subsequence, we have

Px

limq∈Q+q↓t

Xq = Xt

= 1. (1.33)

By our definition of Xt, the limit in (1.33) is equal to Xt on Ω0. Hence, for

all x ∈ D∆, we have Px[Xt = Xt] for each t, implying that X is a version ofX. This then also yields

Ex[e〈u,Xt〉

]= Ex

[e〈u,Xt〉

]and augmentation of (F0

t ) with respect to Px ensures that Xt ∈ Fxt for each

t. We therefore conclude that X is an affine process with respect to (Fxt ).

If D is non-compact, the cadlag version (1.32) on D∆ ∪∞, still denotedby X, can be realized on the space Ω′ := D′(D∆ ∪ ∞) of cadlag pathsω : R+ → D∆ ∪ ∞ with ω(t) = ∆ for t ≥ s, whenever ω(s) = ∆. However,we still have to prove that we can identify ∞ with ∆, as mentioned in Re-mark 1.2.8. In other words, we have to show that ‖ω(t)‖ =∞ for all t ≥ s ifexplosion occurs for some s > 0, that is, ‖ω(s−)‖ =∞. This is stated in theTheorem 1.2.10 below. For its proof let us introduce the following notations:

Due to the convention ‖∆‖ = ∞, we can define the explosion time by(see Cheridito, Filipovic, and Yor [2005] for a similar definition)

Texpl :=

T∆, if T ′k < T∆ for all k,∞, if T ′k = T∆ for some k,

where the stopping times T∆ and T ′k are given by

T∆ := inft > 0 | ‖Xt−‖ =∞ or ‖Xt‖ =∞,T ′k := inft | ‖Xt−‖ ≥ k or ‖Xt‖ ≥ k, k ≥ 1.


Moreover, we denote by relint(C) the relative interior of a set C definedby

relint(C) = x ∈ C |B(x, r) ∩ aff(C) ⊆ C for some r > 0,

where aff(C) denotes the affine hull of C.

Lemma 1.2.9. Let X be an affine process with cadlag paths in D∆ ∪ ∞and let x ∈ D be fixed. If

Px[Texpl <∞] > 0, (1.34)

then relint(ReU) 6= ∅ and we have Px-a.s.

limt↑Texpl

e〈u,Xt〉 = 0

for all u ∈ relint(ReU).

Proof. Let us first establish that under Assumption (1.34), relint ReU 6= ∅.To this end, we denote by Ωexpl the set

Ωexpl = ω ∈ Ω′ | Texpl(ω) <∞.

Then it follows from Proposition 1.2.1 and 1.2.4 that, for Px-almost everyω ∈ Ωexpl, there exist some (T (ω), v(ω)) ∈ (Texpl(ω),∞)× iV such that

limt↑Texpl(ω)

Φ(T (ω)− t, v(ω)) 6= 0

and

limt↑Texpl(ω)

NT (ω),v(ω)t (ω) = lim

t↑Texpl(ω)e〈Ψ(T (ω)−t,v(ω)),Xt(ω)〉 = 0. (1.35)

This implies that

limt↑Texpl(ω)

〈ReΨ(T (ω)− t, v(ω)), Xt(ω)〉 = −∞, (1.36)

and in particular that U 3 ReΨ(T (ω)−Texpl(ω), v(ω)) 6= 0, which proves theclaim, since ReU ⊆ relint ReU .

Furthermore, by (1.36) we have

limt↑Texpl(ω)

‖Π〈ReU〉(Xt(ω))‖ =∞. (1.37)


Define now the vector space W by

W = ReU ∩ (−ReU).

By the definition of U , |〈w,Xt(ω)〉| is bounded for all t ≤ Texpl(ω) and w ∈ W ,which implies that

limt↑Texpl(ω)

‖Π1(Xt(ω))‖ =∞, (1.38)

where Π1 denotes the projection on the orthogonal complement of W in〈ReU〉. As Π1(ReU) is a proper convex cone6 (see, e.g., Bruns and Gube-ladze [2009, Proposition 1.18]), we thus have for all u ∈ relint(Π1(ReU))

limt↑Texpl(ω)

〈u,Xt(ω)〉 = −∞.

Writing relint(ReU) as

relint(ReU) = W + relint(Π1(ReU)),

then yields the assertion, since |〈w,Xt(ω)〉| is bounded for all t ≤ Texpl(ω)and w ∈ W .

Theorem 1.2.10. Let X be an affine process with cadlag paths in D∆∪∞.Then, for every x ∈ D, the following assertion holds Px-a.s.: If

‖Xs−‖ =∞, (1.39)

then ‖Xt‖ = ∞ for all t ≥ s and s ≥ 0. Identifying ∞ with ∆, then yieldsXt = ∆ for all t ≥ s.

Proof. Let x ∈ D be fixed and let u ∈ relint(ReU). Note that by Lemma 1.2.9relint(ReU) 6= ∅ and that Φ(t, u) and Ψ(t, u) are real-valued functions withvalues in R++ and ReU , respectively. Take now some T > 0 and δ > 0 suchthat

Px [T − δ < Texpl ≤ T ] > 0,

and Ψ(t, u) ∈ relint(ReU) for all t < δ. Consider the martingale

MT,ut = Φ(T − t, u)e〈Ψ(T−t,u),Xt〉, t ≤ T,

which is clearly nonnegative and has cadlag paths. Moreover, by the choice ofδ, it follows from Lemma 1.2.9 and the conventions ‖∆‖ = ∞ and f(∆) = 0for any other function that Px-a.s.

MT,us− = 0, s ∈ (T − δ, T ], (1.40)

6A cone is called proper if K ∩ (−K) = 0 (see also Chapter 2).


if and only if ‖Xs−‖ = ∞ for s ∈ (T − δ, T ]. We thus conclude using Rogersand Williams [1994, Theorem II.78.1] that Px-a.s. MT,u

t = 0 for all t ≥ s,which in turn implies that ‖Xt‖ =∞ for all t ≥ s. This allows us to identify∞ with ∆ and we obtain Xt = ∆ for all t ≥ s. Since T was chosen arbitrarily,the assertion follows.

Combining Theorem 1.2.7 and Theorem 1.2.10 and using Assumption (1.1),we thus obtain the following statement:

Corollary 1.2.11. Let X be an affine process relative to (F0t ). Then there

exists a process X such that, for each x ∈ D∆, X is a Px-version of X,which is an affine process relative to (Fxt ), whose paths are cadlag and satisfy

Px-a.s. Xt = ∆ for t ≥ s, whenever ‖Xs−‖ =∞ or ‖Xs‖ =∞.

Remark 1.2.12. We will henceforth always assume that we are using thecadlag version of an affine process, given in Corollary 1.2.11, which we stilldenote by X. Under this assumption X can now be realized on the spaceΩ = D(D∆) of cadlag paths ω : R+ → D∆ with ω(t) = ∆ for t ≥ s, whenever‖ω(s−)‖ =∞ or ‖ω(s)‖ =∞. The canonical realization of an affine processX is then defined by Xt(ω) = ω(t).

1.3 Right-Continuity of the Filtration

Using the existence of a right-continuous version of an affine process, we cannow show that (Fxt ), that is, the augmentation of (F0

t ) with respect to Px, isright-continuous.

Theorem 1.3.1. Let x ∈ D be fixed and let X be an affine process relative to(Fxt ) with cadlag paths. Then (Fxt ) is right-continuous.

Proof. We adapt the proof of Protter [2005, Theorem I.4.31] to our setting.We have to show that for every t ≥ 0, Fxt+ = Fxt , where Fxt+ =

⋂s>tFxt .

Since the filtration is increasing, it suffices to show that Fxt =⋂n≥1Fxt+ 1

n

. In

particular, we only need to prove that

Ex[e〈u1,Xt1 〉+···+〈uk,Xtk 〉

∣∣∣Fxt ] = Ex[e〈u1,Xt1 〉+···+〈uk,Xtn 〉

∣∣∣Fxt+] (1.41)

for all (t1, . . . , tk) and all (u1, . . . , uk) with ti ∈ R+ and ui ∈ U , since thisimplies Ex[Z|Fxt ] = Ex[Z|Fxt+] for every bounded Z ∈ Fx. As both Fxt+ andFxt contain the nullsets N (Fx), this then already yields Fxt+ = Fxt for allt ≥ 0.

1.3. Right-Continuity of the Filtration 29

In order to prove (1.41), let t ≥ 0 be fixed and take first t1 ≤ t2 · · · ≤ tk ≤ t.Then we have for all (u1, . . . , uk)

Ex[e〈u1,Xt1 〉+···+〈uk,Xtk 〉

∣∣∣Fxt ] = Ex[e〈u1,Xt1 〉+···+〈uk,Xtk 〉

∣∣∣Fxt+]= e〈u1,Xt1 〉+···+〈uk,Xtk 〉.

In the case tk > tk−1 · · · > t1 > t, we give the proof for k = 2 for notationalconvenience. Let t2 > t1 > t and fix u1, u2 ∈ U . Then we have by the affineproperty

Ex[e〈u1,Xt1 〉+〈u2,Xt2 〉

∣∣∣Fxt+] = lims↓t

Ex[e〈u1,Xt1 〉+〈u2,Xt2 〉

∣∣∣Fxs ]= lim

s↓tEx[Ex[e〈u1,Xt1 〉+〈u2,Xt2 〉

∣∣∣Fxt1] ∣∣∣Fxs ]= Φ(t2 − t1, u2) lim

s↓tEx[e〈u1+Ψ(t2−t1,u2),Xt1 〉

∣∣∣Fxs ] .If Φ(t2−t1, u2) = 0, it follows by the same step that Ex[e〈u1,Xt1 〉+〈u2,Xt2 〉 | Fxt ] =0, too. Otherwise, we have by Proposition 1.1.6 (ii), Ψ(t2 − t1, u2) ∈ U , andby the definition of U also u1 + Ψ(t2 − t1, u2) ∈ U . Hence, again by the affineproperty and right-continuity of t 7→ Xt(ω), the above becomes

Ex[e〈u1,Xt1 〉+〈u2,Xt2 〉

∣∣∣Fxt+]= Φ(t2 − t1, u2) lim

s↓tΦ(t1 − s, u1 + Ψ(t2 − t1, u2))e〈Ψ(t1−s,u1+Ψ(t2−t1,u2)),Xs〉

= Φ(t2 − t1, u2)Φ(t1 − t, u1 + Ψ(t2 − t1, u2))e〈Ψ(t1−t,u1+Ψ(t2−t1,u2)),Xt〉

= Ex[e〈u1,Xt1 〉+〈u2,Xt2 〉

∣∣∣Fxt ] .This yields (1.41) and by the above arguments we conclude that Fxt+ = Fxtfor all t ≥ 0.

Remark 1.3.2. A consequence of Theorem 1.3.1 is that (Ω,Ft, (Fxt ),Px) sat-isfies the usual conditions, since

(i) Fx is Px-complete,

(ii) Fx0 contains all Px-null-sets in Fx,

(iii) (Fxt ) is right-continuous.

Let us now set

F :=⋂x∈D∆

Fx, Ft :=⋂x∈D∆

Fxt . (1.42)


Then (Ω,F , (Ft),Px) does not necessarily satisfy the usual conditions, butFt = Ft+ still holds true. Moreover, it follows e.g. from Revuz and Yor [1999,Proposition III.2.12, III.2.14] that, for each t, Xt is Ft-measurable and aMarkov process relative to (Ft).

Unless otherwise mentioned, we henceforth always consider affine pro-cesses on the filtered space (Ω,F , (Ft)), where Ω = D(D∆), as described inRemark 1.2.12, and F , Ft are given by (1.42). Notice that these assumptionson the probability space correspond to the standard setting considered for Fellerprocesses (compare Rogers and Williams [1994, Definition III.7.16, III.9.2]).

Similar as in the case of Feller processes, we can now formulate and provethe strong Markov property for affine processes using the above setting andin particular the right-continuity of the sample paths.

Theorem 1.3.3. Let X be an affine process and let T be a (Ft)-stopping time.Then for each bounded Borel measurable function f and s ≥ 0

Ex [f(XT+s)|FT ] = EXT [f(Xs)] , Px-a.s.

Proof. This result can be shown by the same arguments used to prove thestrong Markov property of Feller processes (see, e.g., Rogers and Williams[1994, Theorem 8.3, Theorem 9.4]), namely by using a dyadic approximationof the stopping time T and applying the Markov property. Instead of usingC0-functions and the Feller property, we here consider the family of functionsx 7→ e〈u,x〉 |u ∈ iV and the affine property, which asserts in particular that

x 7→ Ex[e〈u,Xt〉

]= Pte

〈u,x〉 = Φ(t, u)e〈Ψ(t,u),x〉

is continuous. This together with the right-continuity of paths then impliesfor every Λ ∈ FT and u ∈ iV

Ex[e〈u,XT+s〉1Λ

]= Ex

[Pse

〈u,XT 〉1Λ

].

The assertion then follows by the same arguments as in Rogers and Williams[1994, Theorem 8.3] or Chung and Walsh [2005, Theorem 2.3.1].

1.4 Semimartingale Property

We shall now relate affine processes to semimartingales, where, for everyx ∈ D, semimartingales are understood with respect to the filtered proba-bility space (Ω,F , (Ft),Px) defined above. By convention, we call X a semi-martingale if X1[0,T∆) is a semimartingale, where – as a consequence of The-orem 1.2.10 and Corollary 1.2.11 – we can now define the lifetime T∆ by

T∆(ω) = inft > 0 |Xt(ω) = ∆. (1.43)

1.4. Semimartingale Property 31

Let us start with the following definition for general Markov processes(compare Cinlar, Jacod, Protter, and Sharpe [1980, Definition 7.1]):

Definition 1.4.1 (Extendend Generator). An operator G with domain DG iscalled extended generator for a Markov process X (relative to some filtration(Ft)) if DG consists of those Borel measurable functions f : D∆ → C for whichthere exists a function Gf such that the process

f(Xt)− f(x)−∫ t

0

Gf(Xs−)ds

is a well-defined (Ft,Px)-local martingale for every x ∈ D∆.

In the following lemma we consider a particular class of functions for whichit is possible to state the form of the extended generator for a Markov processin terms of its semigroup.

Lemma 1.4.2. Let X be a D∆-valued Markov process relative to some filtra-tion (Ft). Suppose that u ∈ U and η > 0. Consider the function

gu,η : D → C, x 7→ gu,η(x) :=1

η

∫ η

0

Pse〈u,x〉ds.

Then, for every x ∈ D,

Mut := gu,η(Xt)− gu,η(X0)−

∫ t

0

1

η

(Pηe

〈u,Xs−〉 − e〈u,Xs−〉)ds

is a (complex-valued) (Ft,Px)-martingale and thus gu,η(X) is a (complex-valued) special semimartingale.

Proof. Since gu,η and Pηe〈u,·〉 − e〈u,·〉 are bounded, Mu

t is integrable for each tand we have

Ex [Mut |Fr]

= Mur + Ex

[gu,η(Xt)− gu,η(Xr)−

∫ t

r

1

η

(Pηe

〈u,Xs−〉 − e〈u,Xs−〉)ds∣∣∣Fr]

= Mur + EXr

[gu,η(Xt−r)− gu,η(X0)−

∫ t−r

0

1

η

(Pηe

〈u,Xs−〉 − e〈u,Xs−〉)ds

]= Mu

r +1

η

∫ t−r+η

t−rPse

〈u,Xr〉ds− 1

η

∫ η

0

Pse〈u,Xr〉ds

− 1

η

∫ t−r+η

η

Pse〈u,Xr〉ds+

1

η

∫ t−r

0

Pse〈u,Xr〉ds

= Mur .


Hence Mu is a (Ft,Px)-martingale and thus gu,η(X) a special semimartingale,since it is the sum of a martingale and a predictable finite variation process.

Remark 1.4.3. Lemma 1.4.2 asserts that the extended generator applied togu,η is given by Ggu,η(x) = 1

η

(Pηe

〈u,x〉 − e〈u,x〉). Note that for general Markov

processes and even for affine processes we do not know whether the “pointwise”infinitesimal generator applied to

e〈u,x〉 = limη→0

gu,η = limη→0

1

η

∫ η

0

Pse〈u,x〉ds,

that is,

limη→0

1

η

(Pηe

〈u,x〉 − e〈u,x〉),

is well-defined or not.7 For this reason we consider the family of functionsx 7→ gu,η(x) |u ∈ U , η > 0, which exhibits in the case of affine pro-cesses similar properties as x 7→ e〈u,x〉 |u ∈ U (see Remark 1.4.6 (ii) andLemma 1.4.7 below). These properties are introduced in the following defini-tions (compare Cinlar et al. [1980, Definition 7.7, 7.8]).

Definition 1.4.4 (Full Class). A class C of Borel measurable functions fromD to C is said to be a full class if, for all r ∈ N, there exists a finite familyf1, . . . , fN ∈ C and a function h ∈ C2(CN , D) such that

x = h(f1(x), . . . , fN(x)) (1.44)

for all x ∈ D with ‖x‖ ≤ r.

Definition 1.4.5 (Complete Class). Let β ∈ V , γ ∈ S+(V ), where S+(V ) de-notes the positive semidefinite matrices over V , and let F be a positive measureon V , which integrates (‖ξ‖2∧1), satisfies F (0) = 0 and x+supp(F ) ⊆ D∆

for all x ∈ D. Moreover, let χ : V → V denote some truncation function, thatis, χ is bounded and satisfies χ(ξ) = ξ in a neighborhood of 0. A countable

subset of functions C ⊂ C2b (D) is called complete if, for every x ∈ D, the

countable collection of numbers

κ(f(x)) = 〈β,∇f(x)〉+1

2

∑i,j

γijDijf(x)

+

∫V

(f(x+ ξ)− f(x)− 〈∇f(x), χ(ξ)〉)F (dξ), f ∈ C (1.45)

7In the case of affine processes, this would be implied by the differentiability of Φ andΨ with respect to t, which we only prove in Section 1.5 using the results of this paragraph.


completely determines β, γ and F . A class C of Borel measurable functionsfrom D to C is said to be complete class if it contains such a countable set.

Remark 1.4.6. (i) Note that the integral in (1.45) is well-defined for allf ∈ C2

b (D). This is a consequence of the integrability assumption andthe fact that x+ supp(F ) is supposed to lie in D∆ for all x.

(ii) The class of functions

C∗ :=D → C, x 7→ e〈u,x〉

∣∣u ∈ iV

(1.46)

is a full and complete class. Indeed, for every x ∈ D with ‖x‖ ≤ r, wecan find n linearly independent vectors (u1, . . . , un) such that

Im〈ui, x〉 ∈[−π

2,π

2

].

This implies that x is given by

x =(arcsin

(Ime〈u1,x〉

), . . . , arcsin

(Ime〈un,x〉

))(Imu1, . . . , Imun)−1

and proves that C∗ is a full class. Completeness follows by the samearguments as in Jacod and Shiryaev [2003, Lemma II.2.44].

Lemma 1.4.7. Let X be an affine process with Φ and Ψ given in (1.4).Consider the class of functions

C :=

D → C, x 7→ gu,η(x) :=

1

η

∫ η

0

Φ(s, u)e〈Ψ(s,u),x〉ds∣∣u ∈ iV, η > 0

.

(1.47)

Then C is a full and complete class.

Proof. Let (u1, . . . un) ∈ iV be n linearly independent vectors and define afunction fη : D → Cn by fη,i(x) = gui,η(x). Then the Jacobi matrix Jfη(x) isgiven by

1η

∫ η0 Φ(s, u1)e〈Ψ(s,u1),x〉Ψ1(s, u1)ds . . . 1

η

∫ η0 Φ(s, u1)e〈Ψ(s,u1),x〉Ψn(s, u1)ds

.... . .

...1η

∫ η0 Φ(s, un)e〈Ψ(s,un),x〉Ψ1(s, un)ds . . . 1

η

∫ η0 Φ(s, un)e〈Ψ(s,un),x〉Ψn(s, un)ds

.

In particular, the imaginary part of Jfη(x) satisfies

limη→0

ImJfη(x) = (cos(Im〈u1, x〉)Imu1, . . . , cos(Im〈un, x〉)Imun)>, (1.48)


Hence there exists some η > 0 such that the rows of ImJfη are linearly inde-pendent. As Imfη : D → Rn is a C∞(D)-function and as JImfη = ImJfη , itfollows from the inverse function theorem that, for each x0 ∈ D, there existssome r0 > 0 such that Imfη : B(x0, r0) → W has a C∞(W ) inverse, whereW = Imfη(B(x0, r0)).

Let now r ∈ N and consider x ∈ D with ‖x‖ ≤ r. By choosing the linearlyindependent vectors (u1, . . . , un) and η > 0 appropriately, we can guaranteethat r0 ≥ ‖x0‖ + r. Indeed, for every δ > 0, we can choose the linearlyindependent vectors (u1, . . . , un) such that |〈ui, x〉| < δ. Assume now withoutloss of generality that 0 ∈ D and let x0 = 0. Due to (1.48) we can thus assurethat for all x ∈ B(0, r) ∩D

‖ limη→0

J−1Imfη

(0) limη→0

JImfη(x)− I‖

= ‖(Imu1, . . . , Imun)−>(cos(Im〈u1, x〉)Imu1, . . . , cos(Im〈un, x〉)Imun)> − I‖< 1

and by the continuity of the matrix inverse the same holds true for η smallenough. The proof of the inverse function theorem (see, e.g., Howard [1997,Theorem 4.2] or Lang [1993, Lemma XIV.1.3]) thus implies that r0 can bechosen to be r. This proves that C is a full class.

Concerning completeness, note that

κ(gu,η(x)) =1

η

∫ η

0

Φ(s, u)e〈Ψ(s,u),x〉

(〈β,Ψ(s, u)〉+

1

2〈Ψ(s, u)γΨ(s, u)〉

+

∫V

(e〈Ψ(s,u),ξ〉 − 1− 〈Ψ(s, u), χ(ξ)〉

)F (dξ)

)ds. (1.49)

In particular, we have

limη→0

κ(gu,η(x)) = κ(e〈u,x〉)

= e〈u,x〉

(〈β, u〉+

1

2〈u, γu〉+

∫V

(e〈u,ξ〉 − 1− 〈u, χ(ξ)〉

)F (dξ)

). (1.50)

By Jacod and Shiryaev [2003, Lemma II.2.44] or simply as a consequenceof the completeness of the class C∗, as defined in (1.46), the function u 7→κ(e〈u,x〉) admits a unique representation of form (1.50), that is, if κ(e〈·,x〉) also

satisfies (1.50) with (β, γ, F ), then β = β, γ = γ and F = F . This propertycarries over to the class C. Indeed, for every x ∈ D, there exists some η > 0such that β = β, γ = γ and F = F if u 7→ κ(gu,η(x)) also satisfies (1.49) with

(β, γ, F ). This proves that C is a complete class.


In order to establish the semimartingale property of X and to study itscharacteristics, we need to handle explosions and killing. Similar to Cheriditoet al. [2005], we consider again the stopping times T∆ defined in (1.43) and T ′kgiven by

T ′k := inft | ‖Xt−‖ ≥ k or ‖Xt‖ ≥ k, k ≥ 1.

By the convention ‖∆‖ = ∞, T ′k ≤ T∆ for all k ≥ 1. As a transition to ∆occurs either by a jump or by explosion, we additionally define the stoppingtimes:

Tjump =

T∆, if T ′k = T∆ for some k,∞, if T ′k < T∆ for all k,

Texpl =

T∆, if T ′k < T∆ for all k,∞, if T ′k = T∆ for some k,

Tk =

T ′k, if T ′k < T∆,∞, if T ′k = T∆.

(1.51)

Note that Tjump < ∞ ∩ Texpl < ∞ = ∅ and limk→∞ Tk = Texpl withTk < Texpl on Texpl < ∞. Hence Texpl is predictable with announcingsequence Tk ∧ k. In order to turn X into a semimartingale and to get explicitexpressions for the characteristics, we stop X before it explodes, which ispossible, since Texpl is predictable. Note that we cannot stop X before it iskilled, as Tjump is totally inaccessible. For this reason we shall concentrate onthe process (Xτ

t ) := (Xt∧τ ), where τ is a stopping time satisfying 0 < τ <Texpl, which exists by the above argument and the cadlag property of X. SinceX = XT∆ , we have

Xτt = Xt1t<(τ∧T∆) +Xτ∧T∆

1t≥(τ∧T∆)

= Xt1t<(τ∧Tjump) +Xτ∧Tjump1t≥(τ∧Tjump),

which implies that a transition to ∆ can only occur through a jump.

Recall that ∆ is assumed to be an arbitrary point which does not lie inD. We can thus identify ∆ with some point in V \D such that every C2

b (D)-function f can be extended continuously to D∆ with f(∆) = 0. Indeed,without loss of generality we may assume that such a point exists, becauseotherwise we can always embed D∆ in V × R.

Theorem 1.4.8. Let X be an affine process and let τ be a stopping timewith τ < Texpl, where Texpl is defined in (1.51). Then X1[0,T∆) and Xτ aresemimartingales with state space D ∪ 0 and D∆, respectively. Moreover, let(B,C, ν) denote the characteristics of Xτ relative to some truncation function


χ. Then there exists a version of (B,C, ν), which is of the form

Bt,i =

∫ t∧τ

0

bi(Xs−)ds,

Ct,ij =

∫ t∧τ

0

cij(Xs−)ds,

ν(ω; dt, dξ) = K(Xt, dξ)1[0,τ ]dt,

(1.52)

where b : D → V and c : D → S+(V ) are Borel measurable functionsand K(x, dξ) is a positive kernel from (D,D) into (V,B(V )), which satis-fies

∫V

(‖ξ‖2 ∧ 1)K(x, dξ) < ∞, K(x, 0) = 0 and x + supp(K(x, ·)) ⊆ D∆

for all x ∈ D.

Proof. We adapt the proof of Cinlar et al. [1980, Theorem 7.9 (ii), (iii)] to oursetting. By Lemma 1.4.2,

gu,η(X) =1

η

∫ η

0

Φ(s, u)e〈Ψ(s,u),X〉ds

is a semimartingale for every u ∈ U and η > 0. Since Lemma 1.4.7 assertsthat C, as defined in (1.47), is a full class, an application of Ito’s formula to thefunction hi appearing in (1.44) shows that Xi coincides with a semimartingaleon each stochastic interval [0, τr[, where

τr = inft ≥ 0 | ‖Xt‖ ≥ r ∧ T∆.

Since we have Px-a.s. limr→∞ τr = T∆ and since being a semimartingale is alocal property (see Jacod and Shiryaev [2003, Proposition I.4.25]), we concludethat X1[0,T∆) is a semimartingale.

Let now τ denote a stopping time with τ < Texpl. Then Xτ is also a semi-martingale with state space D∆, since explosion is avoided and the transitionto ∆ can only occur via killing, that is, a jump to ∆, which is incorporated inthe jump characteristic (see Cheridito et al. [2005, Section 3]).

By Cinlar et al. [1980, Theorem 6.25], one can find a version of the char-acteristics (B,C, ν) of Xτ , which is of the form

Bt,i =

∫ t∧τ

0

bs−,idAs,

Ct,ij =

∫ t∧τ

0

cs−,ijdAs,

ν(ω; dt, dξ) = 1[0,τ ]dAt(ω)Kω,t(dξ),

(1.53)


where A is an additive process of finite variation, which is Px-indistinguishablefrom a (Ft)-predictable process, b and c are (Ft)-optional processes with values

in V and S+(V ), respectively, and Kω,t(dξ) is a positive kernel from (Ω ×R+,O(Ft))8 into (V,B(V )) satisfying

∫V

(‖ξ‖2∧1)Kω,t(dξ) <∞, Kω,t(0) = 0

and Xt(ω) + supp(Kω,t) ⊆ D∆ for all t ∈ [0, τ ] and Px-almost all ω. Moreover,by Jacod and Shiryaev [2003, Theorem II.2.42], for every f ∈ C2

b (D∆), theprocess

f(Xτt )− f(x)−

∫ t∧τ

0

〈bs−,∇f(Xs−)〉dAs −1

2

∫ t∧τ

0

∑i,j

cs−,ijDijf(Xs−)dAs

−∫ t∧τ

0

∫V

(f(Xs− + ξ)− f(Xs−)− 〈∇f(Xs−), χ(ξ)〉) Kω,s−(dξ)dAs (1.54)

is a (Ft,Px)-local martingale and the last three terms are of finite variation.Note here that ∆ is assumed to be an arbitrary point in V \D such that wecan extend f ∈ C2

b (D) continuously with f(∆) = 0. Let us denote

Lf(Xt−(ω)) := 〈bt−,∇f(Xt−(ω))〉 − 1

2

∑i,j

ct−,ijDijf(Xt−(ω))

−∫V

(f(Xt−(ω) + ξ)− f(Xt−(ω))− 〈∇f(Xt−(ω)), χ(ξ)〉) Kω,t−(dξ). (1.55)

As proved in Lemma 1.4.7, the class of functions C defined in (1.47) is

complete. Let now C ⊂ C be the countable set satisfying the property statedin Definition 1.4.5 and let gη,u ∈ C for some u ∈ iV and η > 0. ThenLemma 1.4.2 and Remark 1.4.1 imply that

gη,u(Xτt )− gη,u(x)−

∫ t∧τ

0

Ggη,u(Xs−)ds

= gη,u(Xτt )− gη,u(x)−

∫ t∧τ

0

1

η

(Pηe

〈u,Xs−〉 − e〈u,Xs−〉)ds (1.56)

is a (Ft,Px)-martingale, while (∫ t∧τ

0Ggη,u(Xs−)ds) is a predictable finite vari-

ation process. Due to (1.54), (1.55) and uniqueness of the canonical decompo-sition of the special semimartingale gη,u(X

τ ) (see Jacod and Shiryaev [2003,Definition I.4.22, Corollary I.3.16]), we thus have∫ t∧τ

0

Lgη,u(Xs−)dAs =

∫ t∧τ

0

Ggη,u(Xs−)ds up to an evanescent set.

(1.57)

8Here, O(Ft) denotes the (Ft)-optional σ-algebra.


Set now

Λ =

(ω, t) : Lgη,u(X(t∧τ∧T∆)−(ω)) = 0 for every gη,u ∈ C.

Then the characteristic property (1.45) of C implies that Λ is exactly the

set where b = 0, c = 0 and K = 0. Hence we may replace A by 1ΛcAwithout altering (1.53), that is, we can suppose that 1ΛA = 0. This propertytogether with (1.57) implies that dAt dt Px-a.s. Hence we know that there

exists a triplet (b′, c′, K ′) such that A replaced by t and (b, c, K) replaced by(b′, c′, K ′) satisfy all the conditions of (1.53). In particular, we have by Jacodand Shiryaev [2003, Proposition II.2.9 (i)] that Xτ is quasi-left continuous.By Cinlar et al. [1980, Theorem 6.27], it thus follows that

b′t = b(Xt)1[0,τ ],

c′t = c(Xt)1[0,τ ],

K ′ω,t(dξ) = K(Xt, dξ)1[0,τ ],

where the functions b, c and the kernel K have the properties stated in (1.52).This proves the assertion.

1.5 Regularity

By means of the above derived semimartingale property, in particular thefact that the characteristics are absolutely continuous with respect to theLebesgue measure, we can prove that every affine process is regular in thefollowing sense:

Definition 1.5.1 (Regularity). An affine process X is called regular if forevery u ∈ U the derivatives

F(u) =∂Φ(t, u)

∂t

∣∣∣∣∣t=0

, R(u) =∂Ψ(t, u)

∂t

∣∣∣∣∣t=0

(1.58)

exist and are continuous on Um for every m ≥ 1.

Remark 1.5.2. In the case of the canonical state space D = Rm+ × Rn−m,

the derivative of φ(t, u) at t = 0 is denoted by F (u) (see Duffie et al. [2003,Equation (3.10)] and Remark 1.1.5). Since Φ(t, u) = eφ(t,u), we have

F(u) = ∂tΦ(t, u)|t=0 = eφ(0,u)∂tφ(t, u)|t=0 = ∂tφ(t, u)|t=0 = F (u).

By the definition of R(u) in Duffie et al. [2003, Equation 3.11], we also haveR(u) = R(u). Hence F and R coincide with F and R, as defined in Duffieet al. [2003].

1.5. Regularity 39

Lemma 1.5.3. Let X be an affine process. Then the functions t 7→ Φ(t, u)and t 7→ Ψi(t, u), i ∈ 1, . . . , n, defined in (1.4) are of finite variation for allu ∈ U .

Proof. Due to Assumption 1.1.3, there exist n+1 vectors such that (x1, . . . , xn)are linearly independent and xn+1 =

∑ni=1 λixi for some λ ∈ V with

∑ni=1 λi 6=

1.Let us now take n+ 1 affine processes X1, . . . , Xn+1 such that

Pxi [X i0 = xi] = 1

for all i ∈ 1, . . . , n + 1. It then follows from Theorem 1.4.8 that, for everyi ∈ 1, . . . , n + 1, X i is a semimartingale with respect to the filtered prob-ability space (Ω,F , (Ft),Pxi). We can then construct a filtered probabilityspace (Ω′,F ′, (F ′t),P′), with respect to which X1, . . . , Xn+1 are independentsemimartingales such that P′ (X i)−1 = Pxi . One possible construction is theproduct probability space (Ωn+1,⊗n+1

i=1 F , (⊗n+1i=1 Ft),⊗n+1

i=1 Pxi).We write yi = (1, xi)

> and Y i = (1, X i)> for i ∈ 1, . . . , n + 1. Thenthe definition of xi implies that (y1, . . . , yn+1) are linearly independent. More-over, as X i exhibits cadlag paths for all i ∈ 1, . . . , n + 1, there exists somestopping time δ > 0 such that, for all ω ∈ Ω′ and t ∈ [0, δ(ω)), the vec-tors (Y 1

t (ω), . . . , Y n+1t (ω)) are also linearly independent. Let now T > 0 and

u ∈ U be fixed and choose some 0 < ε(ω) ≤ δ(ω) such that, for all t ∈ [0, ε(ω)),Φ(T − t, u) 6= 0.

Denoting the (F ′t,P′)-martingales Φ(T − t, u)e〈Ψ(T−t,u),Xit〉 by MT,u,i

t andchoosing the right branch of the complex logarithm, we thus have for allt ∈ [0, ε(ω)) 1 X1

t,1(ω) . . . X1t,n(ω)

......

. . ....

1 Xn+1t,1 (ω) . . . Xn+1

t,n (ω)

−1 lnMT,u,1

t (ω)...

lnMT,u,n+1t (ω)

=

ln Φ(T − t, u)Ψ1(T − t, u)

...Ψn(T − t, u)

.

This implies that (Φ(s, u))s and (Ψ(s, u))s coincide on the stochastic interval(T − ε(ω), T ] with deterministic semimartingales and are thus of finite vari-ation. As this holds true for all T > 0, we conclude that t 7→ Φ(t, u) andt 7→ Ψi(t, u) are of finite variation.

Using Lemma 1.5.3 and Theorem 1.4.8, we are now prepared to prove reg-ularity of affine processes. Additionally, our proof reveals that the functions Fand R have parameterizations of Levy-Khintchine type and that the (differ-ential) semimartingale characteristics introduced in (1.52) depend in an affineway on X.


Theorem 1.5.4. Every affine process is regular. Moreover, the functions Fand R, as defined in (1.58), are of the form

F(u) = 〈u, b〉+1

2〈u, au〉 − c

+

∫V

(e〈u,ξ〉 − 1− 〈u, χ(ξ)〉

)m(dξ), u ∈ U ,

〈R(u), x〉 = 〈u,B(x)〉+1

2〈u,A(x)u〉 − 〈γ, x〉

+

∫V

(e〈u,ξ〉 − 1− 〈u, χ(ξ)〉

)M(x, dξ), u ∈ U ,

where χ : V → V denotes some truncation function such that χ(∆ − x) = 0for all x ∈ D, b ∈ V , a ∈ S(V ), m is a (signed) measure, c ∈ R, γ ∈ V andx 7→ B(x), x 7→ A(x), x 7→ M(x, dξ) are restrictions of R-linear maps on Vsuch that

b(x) = b+B(x),

c(x) = a+ A(x),

K(x, dξ) = m(dξ) +M(x, dξ) + (c+ 〈γ, x〉)δ(∆−x)(dξ).

Here, the left hand side corresponds to the (differential) semimartingale char-acteristics introduced in (1.52).

Furthermore, on the set Q = (t, u) ∈ R+ × U |Φ(s, u) 6= 0, for all s ∈[0, t], the functions Φ and Ψ satisfy the ordinary differential equations

∂tΦ(t, u) = Φ(t, u)F(Ψ(t, u)), Φ(0, u) = 1, (1.59)

∂tΨ(t, u) = R(Ψ(t, u)), Ψ(0, u) = u ∈ U . (1.60)

Remark 1.5.5. Recall that without loss of generality we identify ∆ with somepoint in V \D such that every f ∈ C2

b (D) can be extended continuously to D∆

with f(∆) = 0.

Proof. Let m ≥ 1 and u ∈ Um be fixed and choose Tu > 0 such that Φ(Tu −t, u) 6= 0 for all t ∈ [0, Tu]. As t 7→ Φ(t, u) and t 7→ Ψ(t, u) are of finitevariation by Lemma 1.5.3, their derivatives with respect to t exist almosteverywhere and we can write

Φ(Tu − t, u)− Φ(Tu, u) =

∫ t

0

−dΦ(Tu − s, u),

Ψi(Tu − t, u)−Ψi(Tu, u) =

∫ t

0

−dΨi(Tu − s, u),

1.5. Regularity 41

for i ∈ 1, . . . , n. Moreover, by the semiflow property of Φ and Ψ (seeProposition 1.1.6 (iv)), differentiability of Φ(t, u) and Ψ(t, u) with respect tot at some Tu ≥ ε > 0 implies that the derivatives ∂t|t=0Ψ(t,Ψ(ε, u)) and∂t|t=0Φ(t,Ψ(ε, u)) exist as well. Let now (εk)k∈N denote a sequence of pointswhere Φ(t, u) and Ψ(t, u) are differentiable such that limk→∞ εk = 0. Thenthere exists a sequence (uk)k∈N given by

uk = Ψ(εk, u) ∈ U with limk→∞

uk = u (1.61)

such that the derivatives

∂t|t=0Ψ(t, uk), ∂t|t=0Φ(t, uk) (1.62)

exist for every k ∈ N. Moreover, since |Ex[exp(〈u,Xεk〉)]| < m, there existssome constant M such that uk ∈ UM for all k ∈ N.

Furthermore, due to Theorem 1.4.8, the canonical semimartingale repre-sentation of Xτ (see Jacod and Shiryaev [2003, Theorem II.2.34]), where τ isa stopping time with τ < Texpl, is given by

Xτt = x+

∫ t∧τ

0

b(Xs−)ds+N τt +

∫ t∧τ

0

∫V

(ξ − χ(ξ))µXτ

(ω; ds, dξ),

where µXτ

is the random measure associated with the jumps of Xτ and N τ

is a local martingale, namely the sum of the continuous martingale part andthe purely discontinuous one, that is,

∫ t∧τ

0

∫V

χ(ξ)(µXτ

(ω; ds, dξ)−K(Xs−, dξ)ds).

Applying Ito’s formula (relative to the measure Px) to the martingale MTu,ut∧τ =


Φ(Tu − (t ∧ τ), u)e〈Ψ(Tu−(t∧τ),u),Xt∧τ 〉, we obtain

MTu,ut∧τ = MTu,u

0 +

∫ t∧τ

0

MTu,us−

(−dΦ(Tu − s, u)

Φ(Tu − s, u)+ 〈−dΨ(Tu − s, u), Xs−〉

)+

∫ t∧τ

0

MTu,us−

(〈Ψ(Tu − s, u), b(Xs−)〉

+1

2〈Ψ(Tu − s, u), c(Xs−)Ψ(Tu − s, u)〉

+

∫V

(e〈Ψ(Tu−s,u),ξ〉 − 1− 〈Ψ(Tu − s, u), χ(ξ)〉

)K(Xs−, dξ)

)ds

+

∫ t∧τ

0

MTu,us− 〈Ψ(Tu − s, u), dN τ

s 〉

+

∫ t∧τ

0

∫V

MTu,us−

(e〈Ψ(Tu−s,u),ξ〉 − 1− 〈Ψ(Tu − s, u), χ(ξ)〉

)×(µX

τ

(ω; ds, dξ)−K(Xs−, dξ)ds).

As the last two terms are local martingales and as the rest is of finite variation,we thus have, for almost all t ∈ [0, Tu ∧ τ ], Px-a.s. for every x ∈ D,

dΦ(Tu − t, u)

Φ(Tu − t, u)+ 〈dΨ(Tu − t, u), Xt−〉

= 〈Ψ(Tu − t, u), b(Xt−)〉 dt+1

2〈Ψ(Tu − t, u), c(Xt−)Ψ(Tu − t, u)〉 dt

+

∫V

(e〈Ψ(Tu−t,u),ξ〉 − 1− 〈Ψ(Tu − t, u), χ(ξ)〉

)K(Xt−, dξ)dt.

(1.63)

Note in particular that due to x + supp(K(x, ·)) ⊆ D∆ for every x ∈ D, theabove integral is well-defined. By setting t = Tu on a set of positive measurewith Px[τ ≥ Tu] and letting Tu → 0, we obtain due to (1.62) for all k ∈ N andx ∈ D

∂t|t=0Φ(t, uk) + 〈∂t|t=0Ψ(t, uk), x〉

= 〈uk, b(x)〉 dt+1

2〈uk, c(x)uk〉 dt+

∫V

(e〈uk,ξ〉 − 1− 〈uk, χ(ξ)〉

)K(x, dξ)dt,

(1.64)

where (uk) is given by (1.61). Since the right hand side is continuous in uk,which is again a consequence of the support properties of K(x, ·) and thefact that uk ∈ UM for all k ∈ N, the limit for uk → u of the left hand side

1.5. Regularity 43

exists as well. By the affine independence of the n + 1 elements in D, thecoefficients ∂t|t=0Φ(t, uk) and ∂t|t=0Ψ(t, uk) converge for uk → u, whence thelimit is affine, too. Since u was arbitrary, it follows that

〈u, b(x)〉 dt+1

2〈u, c(x)u〉 dt+

∫V

(e〈u,ξ〉 − 1− 〈u, χ(ξ)〉

)K(x, dξ)dt

is an affine function in x for all u ∈ U .

By uniqueness of the Levy-Khintchine representation and the assumptionthatD contains n+1 affinely independent elements, this implies that x 7→ b(x),x 7→ c(x) and x 7→ K(x, dξ) are affine functions in the following sense:

b(x) = b+B(x),

c(x) = a+ A(x),

K(x, dξ) = m(dξ) +M(x, dξ) + (c+ 〈γ, x〉)δ(∆−x)(dξ),

where b ∈ V , a ∈ S(V ), m a (signed) measure, c ∈ R, γ ∈ V and x 7→ B(x),x 7→ A(x), x 7→ M(x, dξ) are restriction of R-linear maps on V . Indeed,c+ 〈γ, x〉 corresponds to the killing rate of the process, which is incorporatedin the jump measure. Here, we explicitly use the convention e〈u,∆〉 = 0 andthe fact that χ(∆− x) = 0 for all x ∈ D.

Moreover, for t small enough, we now have for all u ∈ U

Φ(t, u)− Φ(0, u) =

∫ t

0

Φ(s, u)

(〈Ψ(s, u), b〉+

1

2〈Ψ(s, u), aΨ(s, u)〉 − c

+

∫V

(e〈Ψ(s,u),ξ〉 − 1− 〈Ψ(s, u), χ(ξ)〉

)m(dξ)

)ds,

〈Ψ(t, u)−Ψ(0, u), x〉 =

∫ t

0

(〈Ψ(s, u), B(x)〉+

1

2〈Ψ(s, u), A(x)Ψ(s, u)〉

− 〈γ, x〉

+

∫V

(e〈Ψ(s,u),ξ〉 − 1− 〈Ψ(s, u), χ(ξ)〉

)M(x, dξ)

)ds.

Note again that the properties of the support of K(x, ·) carry over to themeasures M(x, ·) and m(·) implying that the above integrals are well-defined.Due to the continuity of t 7→ Φ(t, u) and t 7→ Ψ(t, u), we can conclude that thederivatives of Φ and Ψ exist at 0 and are continuous on Um for every m ≥ 1,


since they are given by

F(u) =∂Φ(t, u)

∂t

∣∣∣∣∣t=0

= 〈u, b〉+1

2〈u, au〉 − c

+

∫V

(e〈u,ξ〉 − 1− 〈u, χ(ξ)〉

)m(dξ),

〈R(u), x〉 =

⟨∂Ψ(t, u)

∂t

∣∣∣∣∣t=0

, x

⟩= 〈u,B(x)〉+

1

2〈u,A(x)u〉 − 〈γ, x〉

+

∫V

(e〈u,ξ〉 − 1− 〈u, χ(ξ)〉

)M(x, dξ).

This proves the first part of the theorem.By the regularity of X, we are now allowed to differentiate the semiflow

equations (1.5) on the set Q = (t, u) ∈ R+×U |Φ(s, u) 6= 0, for all s ∈ [0, t]with respect to s and evaluate them at s = 0. As a consequence, Φ and Ψsatisfy (1.59) and (1.60).

Remark 1.5.6. The differential equations (1.59) and (1.60) are called gen-eralized Riccati equations, which is due to the particular form of F and R.

Chapter 2

Affine Processes on ProperConvex Cones

In order to be able to further characterize the functions F and R, as introducedin (1.58), and to show that the generalized Riccati equations admit uniqueglobal solutions, which is needed to establish existence of affine processes, weassume in this chapter the state space D to be a closed proper convex cone.We denote it by K to emphasize the difference to the case of a general statespace. This setting then allows us to prove existence of affine pure jumpprocesses, which is elaborated in Section 2.5. The question of existence ofaffine diffusion processes is treated in Chapter 3, where we suppose furtherstructural assumptions on the state space by building on the framework ofirreducible symmetric cones. Clearly, the case Rn

+ studied in Duffie et al.[2003] is an example of a proper convex cone.1

Most proofs in this chapter are generalizations of the corresponding re-sults on the cone of positive semidefinite matrices obtained in Cuchiero et al.[2011a].

2.1 Definition of Cone-valued Affine Processes

Following Faraut and Koranyi [1994], we call a cone proper if K∩(−K) = 0.We further suppose K to be generating, that is, K contains a basis, which isequivalent to V = K −K. The closed dual cone of K is defined by

K∗ = u ∈ V | 〈x, u〉 ≥ 0 for all x ∈ K.

Note that the above assumptions imply in particular that K∗ is also generatingand proper (see, e.g., Faraut and Koranyi [1994, Proposition I.1.4]). We denote

1Let us remark that Rn+ is actually a reducible symmetric cone (see Chapter 3 for details).

45

46 Chapter 2. Affine Processes on Proper Convex Cones

the open dual cone of K by

K∗ = u ∈ V | 〈x, u〉 > 0 for all x ∈ K,

and by∂K∗ = u ∈ V | 〈x, u〉 = 0 for some x ∈ K

the boundary of K∗. Like any cone, K∗ induces a partial and strict orderrelation on V : for u, v ∈ V , we write u v if v − u ∈ K∗ and u ≺ v ifv − u ∈ K∗. As in Chapter 1, we adjoin to the state space K a point ∆ /∈ Kand set K∆ = K ∪ ∆.

Since the set U , as defined in (1.3), certainly contains −K∗, we can slightlymodify the definition of an affine process, by requiring the affine property onlyfor u ∈ −K∗. Thus, instead of the Fourier-Laplace transform, we here onlyconsider the Laplace transform of X, implying that

Ex[e〈u,Xt〉

]= Φ(t, u)e〈Ψ(t,u),x〉

is real-valued and cannot become 0 for u ∈ −K∗. We thus adjust the definitionof an affine process on K as follows:

Definition 2.1.1 (Cone-valued affine process). A time-homogeneous Markovprocess X relative to some filtration (Ft) and with state space K (augmentedby ∆) is called affine if

(i) it is stochastically continuous, that is, lims→t ps(x, ·) = pt(x, ·) weakly onK for every t ≥ 0 and x ∈ K, and

(ii) its Laplace transform has exponential-affine dependence on the initialstate. This means that there exist functions φ : R+ × K∗ → R andψ : R+ ×K∗ → V such that

Ex[e−〈u,Xt〉

]= Pte

−〈u,x〉 =

∫K

e−〈u,ξ〉pt(x, dξ) = e−φ(t,u)−〈ψ(t,u),x〉, (2.1)

for all x ∈ K and (t, u) ∈ R+ ×K∗.

Remark 2.1.2. (i) Let us remark that the above definition and Defini-tion 1.1.4 are equivalent in the case D = K. Indeed, if X is an affineprocess with state space D = K in the sense of Definition 1.1.4, thenit is clearly also an affine process in the sense of Definition 2.1.1, sincethe only difference is the restriction of U to −K∗.Note that for u ∈ K∗ we have

Φ(t,−u) = e−φ(t,u) and Ψ(t,−u) = −ψ(t, u).

2.1. Definition of Cone-valued Affine Processes 47

For the other direction we have to show that (2.1) can be extended to U ,where it takes the form

Ex[e〈u,Xt〉

]= Φ(t, u)e〈Ψ(t,u),x〉.

Note that U = −K∗ + iV , since K is a cone. Following the proofof Keller-Ressel et al. [2010, Lemma 2.5], let us define the function

g(t, u, x) = Ex[e〈u,Xt〉

]= Pte

〈u,x〉

for (t, u, x) ∈ R+ × U ×K. Due to (2.1), we have

g(t, u, x)g(t, u, y) = g(t, u, x+ y)g(t, u, 0) (2.2)

for all (t, u) ∈ R+ × −K∗ and x, y ∈ K. Now it follows from well-known properties of the Fourier-Laplace transform (see, e.g., Duffie et al.[2003, Lemma A.2]) that both sides are analytic on U = −K∗ + iV .Moreover, they coincide on a set of uniqueness, namely −K∗, since K∗

is generating. By the continuity of the Fourier-Laplace transform in u,equality (2.2) therefore holds on all of U .

Assume now that g(t, u, 0) = 0 for some (t, u) ∈ R+×U , then it followsfrom (2.2) that g(t, u, x) = Ex

[e〈u,Xt〉

]= 0 for all x ∈ K. On the other

hand the setO = (t, u) ∈ R+ × U | g(t, u, 0) 6= 0

is open, which is a consequence of the joint continuity of (t, u) 7→ g(t, u, x),and we can define

h(x) =g(t, u, x)

g(t, u, 0)

for all (t, u) ∈ R+ × O. The function h is measurable and satisfiesh(0) 6= 0 and h(x)h(y) = h(x + y) for all x, y ∈ K. Using a standardresult on measurable solutions of the Cauchy equation (see, e.g., Aczel[1966, Section 2.2]), we conclude that there exists a unique continuousextension of −ψ(t,−u) from −K∗ to U such that

g(t, u, x)

g(t, u, 0)= e〈Ψ(t,u),x〉.

Setting Φ(t, u) = g(t, u, 0) completes the proof.

Note that the arguments allowing to extend the affine property from −K∗to U rely on the cone structure of K, as we need (2.2) to hold for suffi-ciently many x, y ∈ V .


(ii) Due to the equivalence of Definition 1.1.4 and Definition 2.1.1, we canconsider affine processes on proper convex cones on the filtered space(Ω,F , (Ft)), where Ω = D(K∆), as described in Remark 1.2.12, andwhere F ,Ft are given in (1.42). This assumption however only plays arole in Theorem 3.7.2 of Section 3.7.

Similar to Proposition 1.1.6, the following properties of φ and ψ are im-mediate consequences of Definition 2.1.1.

Proposition 2.1.3. Let X be an affine process on K. Then the functions φand ψ satisfy the following properties.

(i) φ maps R+ ×K∗ into R+ and ψ maps R+ ×K∗ into K∗.

(ii) φ and ψ satisfy the semiflow property, that is, for any s, t ≥ 0 andu ∈ K∗ we have

φ(t+ s, u) = φ(t, u) + φ(s, ψ(t, u)), φ(0, u) = 0, (2.3)

ψ(t+ s, u) = ψ(s, ψ(t, u)), ψ(0, u) = u. (2.4)

(iii) φ and ψ are jointly continuous on R+ ×K∗. Furthermore, u 7→ φ(t, u)and u 7→ ψ(t, u) are analytic on K∗.

(iv) For any t ≥ 0 and u, v ∈ K∗ with u v the order relations

φ(t, u) ≤ φ(t, v) and ψ(t, u) ψ(t, v)

hold true.

Proof. The left hand side of (2.1) is clearly bounded by 1 for all x ∈ K.Inserting first x = 0 shows that φ(t, u) can only take values in R+. Forarbitrary x ∈ K the right hand side remains bounded only if ψ(t, u) ∈ K∗,which shows (i).

Assertion (ii) follows from the law of iterated expectations as in Proposi-tion 1.1.6 (iv) or equivalently from the Chapman-Kolmogorov equation, thatis,

e−φ(t+s,u)−〈ψ(t+s,u),x〉 =

∫K

e−〈u,ξ〉pt+s(x, dξ)

=

∫K

ps(x, dξ)

∫K

e−〈u,ξ〉pt(ξ, dξ)

= e−φ(t,u)

∫K

e−〈ψ(t,u),ξ〉ps(x, dξ)

= e−φ(t,u)−φ(s,ψ(t,u))−〈ψ(s,ψ(t,u)),x〉.

2.2. Feller Property and Regularity 49

Taking logarithms and using the fact that K is generating, yields (ii).Joint continuity of φ and ψ can be shown by the same arguments used to

prove Proposition 1.1.6 (i). The second assertion is a consequence of analyt-icity properties of the Laplace transform.

Concerning (iv), let u v, which is equivalent to 〈u, x〉 ≤ 〈v, x〉 for allx ∈ K. Hence, for all t ≥ 0 and x ∈ K, we have

e−φ(t,u)−〈ψ(t,u),x〉 =

∫K

e−〈u,ξ〉pt(x, dξ) ≥∫K

e−〈v,ξ〉pt(x, dξ) = e−φ(t,v)−〈ψ(t,v),x〉,

which yields (iv).

2.2 Feller Property and Regularity

The assumption of a cone state space allows us to prove that (Pt) is a Fellersemigroup on C0(K). In order to show this property, we shall mainly relyon Lemma 2.2.1 below. In addition, this result also enables us to give analternative proof of Theorem 1.5.4, showing regularity on −K∗ for every affineprocess on K. Indeed, regularity for affine processes on cone state spacescan be obtained by arguing as in Keller-Ressel et al. [2010], who obtainedthe corresponding statements on the canonical state space D = Rm

+ × Rn−m

(see Keller-Ressel et al. [2010, Theorem 4.3] and also the PhD thesis of Keller-Ressel [2009]). Let us remark that Theorem 1.5.4 can of course be appliedhere, but since the assumption of a cone state space allows for a simplerproof, we briefly reformulate the arguments of Keller-Ressel et al. [2010] forour setting.

Lemma 2.2.1. Let ψ : R+ ×K∗ → V be any map satisfying ψ(0, u) = u andthe properties (i)–(iv) of Proposition 2.1.3 (regarding the function ψ). Thenψ(t, u) ∈ K∗ for all (t, u) ∈ R+ × K∗.

Proof. We adapt the proof of Keller-Ressel [2009, Proposition 1.10] to oursetting. Assume by contradiction that there exists some (t, u) ∈ R+ × K∗

such that ψ(t, u) ∈ ∂K∗. We show that in this case also ψ( t2, u) ∈ ∂K∗. First

note that

ψ

(t

2, v

) ψ

(t

2, ψ

(t

2, u

))= ψ(t, u) (2.5)

for all v ∈ Θ := v ∈ K∗ : v ψ( t2, u) by Proposition 2.1.3 (ii) and (iv). Take

now some x 6= 0 ∈ K such that 〈x, ψ(t, u)〉 = 0. By (2.5) also 〈x, ψ( t2, v)〉 = 0

for all v ∈ Θ. If ψ( t2, u) ∈ K∗, then Θ is a set with non-empty interior. By


analyticity of ψ, it then follows that 〈x, ψ( t2, u)〉 = 0 for all u ∈ K∗, and hence

ψ( t2, u) ∈ ∂K∗, which is a contradiction. We conclude that ψ( t

2, u) ∈ ∂K∗.

Repeating these arguments yields, for each n ∈ N, the existence of anelement xn 6= 0 ∈ K, for which⟨

xn, ψ

(t

2n, u

)⟩= 0.

Without loss of generality we may assume that ‖xn‖ = 1 for each n. Sincethe unit sphere is compact in finite dimensions, there exists a subsequence nksuch that xnk → x∗ 6= 0, as k → ∞. From the continuity of the functiont 7→ ψ(t, u) and the scalar product we deduce that

0 = limk→∞

⟨xnk , ψ

(t

2nk, u

)⟩= 〈x∗, ψ(0, u)〉 = 〈x∗, u〉 > 0,

which is the desired contradiction.

It is now a direct consequence of this lemma that any affine process X onK is a Feller process.

Theorem 2.2.2. Let X be an affine process on K. Then X is a Feller process.

Proof. By Revuz and Yor [1999, Proposition III.2.4], it suffices to show thatfor all f ∈ C0(K)

limt↓0

Ptf(x) = f(x), for all x ∈ K, (2.6)

Ptf ∈ C0(K), for all t ≥ 0. (2.7)

Property (2.6) is a consequence of stochastic continuity, which implies forall f ∈ C0(K) and x ∈ K

limt↓0

Ptf(x) = f(x).

Concerning (2.7), it suffices to verify this property for a dense subset ofC0(K). By a locally compact version of Stone-Weierstrass’ Theorem (see,

e.g., Semadeni [1971]), the linear span of the sete−〈u,x〉 |u ∈ K∗

is dense

in C0(K). Indeed, it is a subalgebra of C0(K), separates points and van-ishes nowhere, as all elements are strictly positive functions on K. FromLemma 2.2.1 we can deduce that Pte

−〈u,x〉 ∈ C0(K) if u ∈ K∗, since ψ(t, u) ∈K∗ and 〈ψ(t, u), x〉 > 0 for x 6= 0 implying that

Pte−〈u,x〉 = e−φ(t,u)−〈ψ(t,u),x〉

goes to 0 as x→ ∆, whence the assertion is proved.

2.2. Feller Property and Regularity 51

As already mentioned before, Lemma (2.2.1) now allows us to prove reg-ularity of affine processes on K using a different approach than the one ofSection 1.5. In order to avoid confusion concerning the different notations, letus give again a rigorous definition of regularity for an affine process on K.

Definition 2.2.3 (Regularity for cone-valued affine processes). An affine pro-cess X on K is called regular if for all u ∈ K∗ the derivatives

F (u) =∂φ(t, u)

∂t

∣∣∣∣∣t=0

, R(u) =∂ψ(t, u)

∂t

∣∣∣∣∣t=0

(2.8)

exist and are continuous in u.

Remark 2.2.4. Note that in the terminology of Definition 1.5.1, we hererequire regularity on −K∗ instead of U . Moreover, we have

F(−u) = ∂tΦ(t,−u)|t=0 = −e−φ(0,u)∂tφ(t, u)|t=0 = −∂tφ(t, u)|t=0 = −F (u),

R(−u) = ∂tΨ(t,−u)|t=0 = −∂tψ(t, u)|t=0 = −R(u).

Let us now reformulate parts of Theorem 1.5.4 in the context of cone statespaces:

Theorem 2.2.5. Let X be an affine process on K. Then X is regular inthe sense of Definition 2.2.3 and the functions φ and ψ satisfy the ordinarydifferential equations

∂tφ(t, u) = F (ψ(t, u)), φ(0, u) = 0, (2.9)

∂tψ(t, u) = R(ψ(t, u)), ψ(0, u) = u ∈ K∗. (2.10)

Proof. A proof of the above theorem can be obtained by following the linesof Keller-Ressel et al. [2010, Proof of Theorem 4.3]. Observe first that Propo-sition 2.1.3 (iii) implies differentiability of φ(t, u) and ψ(t, u) in u on K∗.Arguing similarly as in the proof of Keller-Ressel et al. [2010, Theorem 4.3]and using Lemma 2.2.1, yields differentiability of φ(t, u) and ψ(t, u) at t = 0and continuity of the derivatives in u for all u ∈ K∗.

Similarly as in the proof of Theorem 1.5.4, we are now allowed to differ-entiate the semiflow equations (2.3) and (2.4) with respect to s and evaluatethem at s = 0. As a consequence, φ and ψ satisfy (2.9) and (2.10).

Remark 2.2.6. Note that in contrast to the general state space (2.9) and (2.10)hold for all (t, u) ∈ R+ ×K∗.


2.3 Necessary Conditions

In this section, we focus on the specific form of the functions F and R, de-fined in (2.8). As already proved in Theorem 1.5.4 for the case of generalstate spaces, F and R have parameterizations of Levy-Khintchine type. InProposition 2.3.2 below, we give an alternative proof of this result, which sub-sequently allows us to establish certain necessary admissibility conditions onthe involved parameters. Hence, in contrast to general state spaces consid-ered in Chapter 1, we can here precisely describe F and R by means of acertain parameter set (see Proposition 2.3.3 below). Moreover, we relate theparticular form of F and R to the notion of quasi-monotonicity.

For the proof of the main results of this section we first provide a conver-gence result for Fourier-Laplace transforms.

Lemma 2.3.1. Let (νn)n∈N be a sequence of measures on V with

Ln(u) =

∫V

e−〈u,ξ〉νn(dξ) <∞ and limn→∞

Ln(u) = L(u), u ∈ K∗,

pointwise, for some finite function L on K∗, continuous at u = 0. Thenνn converges weakly to some finite measure ν on V and the Fourier-Laplacetransform converges for u ∈ K∗ ∪ 0 and v ∈ V to the Fourier-Laplacetransform of ν, that is,

limn→∞

∫V

e−〈u+i v,ξ〉νn(dξ) =

∫V

e−〈u+i v,ξ〉ν(dξ).

In particular, ν(V ) = limn→∞ νn(V ) and

L(u) =

∫V

e−〈u,ξ〉ν(dξ),

for all u ∈ K∗ ∪ 0.

Proof. Since νn(V ) = Ln(0) is bounded, we know that νn has a vague accu-mulation point ν, which is a finite measure on V .

Since Ln(u) < ∞ on K∗, it follows by well-known regularity propertiesof Laplace transforms (see, e.g., Duffie et al. [2003, Lemma A.2]) that thefunctions Ln admit an analytic extension to the strip K∗ + iV , still denotedby Ln:

(u+ i v) 7→ Ln(u+ i v) =

∫V

e−〈u+i v,ξ〉νn(dξ).

Moreover, pointwise convergence of the finite convex functions Ln to L onK∗ implies that this convergence is in fact uniform on compact subsets of

2.3. Necessary Conditions 53

K∗ (see, e.g., Rockafellar [1997, Theorem 10.8]). Hence the functions Ln areuniformly bounded on compact subsets of K∗ and since |Ln(u+ i v)| ≤ Ln(u),also on compact subsets of K∗ + iV . Moreover, since K∗ is generating, K∗

is a set of uniqueness in K∗ + iV . It therefore follows from Vitali’s theorem(see Narasimhan [1971, Chapter 1, Proposition 7]) that the analytic functionsLn converge uniformly on compact subsets of K∗ + iV to an analytic limitthereon. By Levy’s continuity theorem, we therefore know that for any u ∈ K∗the finite measures exp(−〈u, ξ〉)νn(dξ) converge weakly to a limit, which byuniqueness of the weak limit has to equal exp(−〈u, ξ〉)ν(dξ). Whence theonly vague accumulation point of νn is ν. Vague convergence implies weakconvergence if mass is conserved. Continuity of L(u) at u = 0 implies thismass conservation. Indeed, by weak convergence of e−〈εu,ξ〉νn for u ∈ K∗ andε > 0, we arrive at

L(εu) = limn→∞

∫V

e−〈εu,ξ〉νn(dξ) =

∫V

e−〈εu,ξ〉ν(dξ)

=

∫V

e−〈εu,ξ〉1〈u,ξ〉≤0ν(dξ) +

∫V

e−〈εu,ξ〉1〈u,ξ〉>0ν(dξ).

By dominated convergence we thus obtain for ε→ 0

L(0) =

∫V

ν(dξ),

which is the desired mass conservation. Hence we have weak convergenceof the measures νn, which means in turn convergence of the Fourier-Laplacetransform on K∗ ∪ 0+ iV .

2.3.1 Levy Khintchine Form of F and R

We are now prepared to provide the particular functional form of F and R foraffine processes on cones. Due to this assumption on the state space, we arenow able to state more specific restrictions on the involved parameters. As inChapter 1, χ : V → V denotes some bounded continuous truncation functionwith χ(ξ) = ξ in a neighborhood of 0.

Proposition 2.3.2. Let X be an affine process on K. Then the functions Fand R as defined in (2.8) are of the form

F (u) = 〈b, u〉+ c−∫K

(e−〈u,ξ〉 − 1

)m(dξ), (2.11)

R(u) = −1

2Q(u, u) +B>(u) + γ −

∫K

(e−〈u,ξ〉 − 1 + 〈χ(ξ), u〉

)µ(dξ), (2.12)

where


(i) b ∈ K,

(ii) c ∈ R+,

(iii) m is a Borel measure on K satisfying m(0) = 0 and∫K

(‖ξ‖ ∧ 1)m(dξ) <∞.

(iv) Q : V × V → V is a symmetric bilinear function with Q(v, v) ∈ K∗ forall v ∈ V ,

(v) B> : V → V is a linear map,

(vi) γ ∈ K∗,

(vii) µ is a K∗-valued σ-finite Borel measure on K satisfying µ(0) = 0 and∫K

(‖ξ‖2 ∧ 1

)〈x, µ(dξ)〉 <∞ for all x ∈ K.

Proof. In order to derive the particular form of F and R with the aboveparameter restrictions, we follow the approach of Keller-Ressel [2009, Theorem2.6]. Note that the t-derivative of Pte

−〈u,x〉 at t = 0 exists for all x ∈ K andu ∈ K∗, since

limt↓0

Pte−〈u,x〉 − e−〈u,x〉

t= lim

t↓0

e−φ(t,u)−〈ψ(t,u),x〉 − e−〈u,x〉

t

= (−F (u)− 〈R(u), x〉)e−〈u,x〉(2.13)

is well-defined by Theorem 2.2.5. Moreover, we can also write

−F (u)− 〈R(u), x〉 = limt↓0

Pte−〈u,x〉 − e−〈u,x〉

te−〈u,x〉

= limt↓0

1

t

(∫K

e−〈u,ξ−x〉pt(x, dξ)− 1

)= lim

t↓0

(1

t

∫K−x

(e−〈u,ξ〉 − 1

)pt(x, dξ + x) +

pt(x,K)− 1

t

).

By the above equalities and the fact that pt(x,K) ≤ 1, we then obtain foru = 0

0 ≥ limt↓0

pt(x,K)− 1

t= −F (0)− 〈R(0), x〉.


Setting F (0) = c and R(0) = γ yields c ∈ R+ and γ ∈ K∗, hence (ii) and (vi).We thus obtain

−(F (u)− c)− 〈R(u)− γ, x〉 = limt↓0

1

t

∫K−x

(e−〈u,ξ〉 − 1

)pt(x, dξ + x). (2.14)

For every fixed t > 0, the right hand side of (2.14) is the logarithm of theLaplace transform of a compound Poisson distribution supported on K−R+xwith intensity pt(x,K)/t and compounding distribution pt(x, dξ+x)/pt(x,K).Concerning the support, note that the compounding distribution is concen-trated on K − x, which implies that the compound Poisson distribution hassupport on the convex cone K−R+x. By Lemma 2.3.1, the pointwise conver-gence of (2.14) for t→ 0 to some function being continuous at 0 implies weakconvergence of the compound Poisson distributions to some infinitely divisibleprobability distribution ν(x, dy) supported on K − R+x. Indeed, this followsfrom the fact that any compound Poisson distribution is infinitely divisible andthe class of infinitely divisible distributions is closed under weak convergence(see Sato [1999, Lemma 7.8]). Again, by Lemma 2.3.1, the Laplace transformof ν(x, dy) is then given as exponential of the left hand side of (2.14).

In particular, for x = 0, ν(0, dy) is an infinitely divisible distribution withsupport on the cone K. By the Levy–Khintchine formula on proper cones(see, e.g., Skorohod [1991, Theorem 3.21]), its Laplace transform is thereforeof the form

exp

(−〈b, u〉+

∫K

(e−〈u,ξ〉 − 1)m(dξ)

),

where b ∈ K and m is a Borel measure supported on K with m(0) = 0 suchthat ∫

K

(‖ξ‖ ∧ 1)m(dξ) <∞,

yielding (iii). Therefore,

F (u) = 〈b, u〉+ c−∫K

(e−〈u,ξ〉 − 1)m(dξ).

We next obtain the particular form of R. Observe that for each x ∈ Kand k ∈ N,

exp (−(F (u)− c)/k − 〈R(u)− γ, x〉)

is the Laplace transform of the infinitely divisible distribution ν(kx, dy)∗1k ,

where ∗ 1k

denotes the 1k

convolution power. For k →∞, these Laplace trans-forms obviously converge to exp(−〈R(u)− γ, x〉) pointwise in u. Using again


the same arguments as before (an application of Lemma 2.3.1 as below equa-

tion (2.14)), we can deduce that ν(kx, dy)∗1k converges weakly to some in-

finitely divisible distribution L(x, dy) on K − R+x with Laplace transformexp(−〈R(u)− γ, x〉) for u ∈ K∗.

By the Levy-Khintchine formula on V (see Sato [1999, Theorem 8.1]), thecharacteristic function of L(x, dy) has the form

L(x, u) = exp

(1

2〈u,A(x)u〉+ 〈B(x), u〉

+

∫V

(e〈u,ξ〉 − 1− 〈χ (ξ) , u〉

)M (x, dξ)

), (2.15)

for u ∈ iV , where, for every x ∈ D, A(x) ∈ S+(V ) is a symmetric positivesemidefinite linear operator on V , B(x) ∈ V , M(x, ·) a Borel measure on Vsatisfying M(x, 0) ∫

V

(‖ξ‖2 ∧ 1)M(x, dξ) <∞,

and χ some appropriate truncation function. Furthermore, by Sato [1999,Theorem 8.7],∫

V

f(ξ)1

tpt(x, dξ + x)

t→0−→∫V

f(ξ)m(dξ) +

∫V

f(ξ)M(x, dξ) (2.16)

holds true for all f : V → R which are bounded, continuous and vanishing ona neighborhood of 0. We thus conclude that M(x, dξ) has support in K − x.

Therefore, the characteristic function L(x, u) admits an analytic extension toK∗ + iV , which then has to coincide with the Laplace transform for u ∈ K∗.Hence, for all x ∈ K,

− 〈R(u)− γ, x〉 =1

2〈u,A(x)u〉 − 〈B(x), u〉

+

∫V

(e−〈u,ξ〉 − 1 + 〈χ (ξ) , u〉

)M (x, dξ) , u ∈ K∗. (2.17)

As the left side of (2.17) is linear in the components of x and as K is generat-ing, it follows that x 7→ A(x), x 7→ B(x) as well as x 7→

∫E

(‖ξ‖2 ∧ 1)M(x, dξ)for every E ∈ B(V ) are restrictions of linear maps on V . In particular, Con-dition (v) follows immediately. Moreover, 〈u,A(x)v〉 can be written as

〈u,A(x)v〉 = 〈x,Q(u, v)〉, (2.18)


where Q : V ×V → V is a symmetric bilinear function satisfying Q(v, v) ∈ K∗for all v ∈ V , since A(x) is a positive semidefinite operator. This thereforeyields (iv). Similarly, we have for all E ∈ B(V )∫

E

(‖ξ‖2 ∧ 1)M(x, dξ) =

∫E

(‖ξ‖2 ∧ 1)〈x, µ(dξ)〉,

where µ is a K∗-valued Borel measure on V , satisfying µ(0) = 0 and∫V

(‖ξ‖2 ∧ 1

)〈x, µ(dξ)〉 <∞, for all x ∈ K.

Hence it only remains to prove that supp(µ) ⊆ K. In (2.16) take now x = 1ny

for some y ∈ K with ‖y‖ = 1 and nonnegative functions f = fn ∈ Cb(V )with fn = 0 on K − 1

ny. Then, for each n, the left side of (2.16) is zero, since

pt(1ny, ·) is concentrated on K − 1

ny. As supp(m) ⊆ K, the first integral on

the right vanishes as well. Hence

0 =

∫V

fn(ξ)M

(1

ny, dξ

)=

∫V

fn(ξ)

⟨1

ny, µ(dξ)

⟩

for any nonnegative function fn ∈ Cb(V ) with fn = 0 on K − 1ny implies that

supp(µ) ⊆ K− 1ny for each n. Thus we can conclude that supp(µ) ⊆ K, which

proves (vii). Due to the definition of Q and µ together with (2.17), R(u) isclearly of form (2.12).

2.3.2 Parameter Restrictions

In the following we continue the analysis of the function R and derive furtherrestrictions on the involved parameters Q, B> and µ.

Proposition 2.3.3. Let X be an affine process on K with R of form (2.12) forsome Q, B>, γ and µ satisfying the conditions of Proposition 2.3.2 (iv)-(vii).Then, for any u ∈ K∗ and x ∈ K with 〈u, x〉 = 0, we have

(i) 〈x,Q(u, v)〉 = 0 for all v ∈ V ,

(ii)∫K〈χ(ξ), u〉〈x, µ(dξ)〉 <∞,

(iii) 〈x,B>(u)〉 −∫K〈χ(ξ), u〉〈x, µ(dξ)〉 ≥ 0.


Proof. Let u ∈ K∗ and x ∈ K with 〈u, x〉 = 0 be fixed. Define the linearmap U : V → R, v 7→ 〈u, v〉. As established in the proof of Proposition 2.3.2,−〈R(u) − γ, x〉 is the Laplace transform of an infinitely divisible distribu-tion L(x, dy) supported on K − R+x. Similar to (2.17), we denote the Levytriplet of L(x, dy) by (A(x), B(x),M(x, dξ)). Let now Yx be a random vari-able with distribution L(x, dy). Then the distribution of U(Yx) = 〈u, Yx〉,which we denote by Lu(x, dy), is again infinitely divisible and supported onR+. From Sato [1999, Proposition 11.10] we then infer that the Levy triplet(au(x), bu(x), νu(x, dξ)) of Lu(x, dy) with respect to some truncation functionχ on R is given by

au(x) = 〈u,A(x)u〉,

bu(x) = 〈B(x), u〉+

∫K

(χ(〈u, ξ〉)− 〈χ(ξ), u〉)U∗M(x, dξ),

νu(x, dξ) = U∗M(x, dξ).

By the Levy Khintchine formula on R+, we conclude that au(x) = 0, bu(x) ≥ 0and

∫K

(‖ξ‖ ∧ 1)U∗M(x, dξ) <∞. The last condition already implies (ii) andallows to choose χ = 0. Moreover, bu(x) ≥ 0 yields (iii). From

0 = au(x) = 〈u,A(x)u〉 =⟨√

A(x)u,√A(x)u

⟩it follows that 〈v, A(x)u〉 = 0 for all v ∈ V . Hence relation (2.18) implies (i).

Remark 2.3.4. For an interpretation of the conditions of Proposition 2.3.3we refer to Section 3.3.

2.3.3 Quasi-monotonicity

Quasi-monotonicity plays a crucial role in comparison theorems for ordinarydifferential equations and thus appears naturally in the setting of affine pro-cesses. As we shall see in Section 2.4, it is needed to establish global exis-tence and uniqueness for the ordinary differential equations defined in (2.9)and (2.10). In the following we prove that the function R, as given in (2.12),is quasi-monotone increasing if the conditions of Proposition 2.3.3 (i)-(iii) aresatisfied.

Definition 2.3.5 (Quasi-monotonicity). Let U be a subset of V . A functionf : U → V is called quasi-monotone increasing (with respect to K∗ and theinduced order ) if, for all u, v ∈ U and x ∈ K satisfying u v and 〈u, x〉 =〈v, x〉,

〈f(u), x〉 ≤ 〈f(v), x〉.


Accordingly, we call f quasi-constant if both f and −f are quasi-monotoneincreasing.

Remark 2.3.6. Note that in a one-dimensional vector space any function isquasi-monotone. It is only in dimension greater than one that the notion ofquasi-monotonicity becomes meaningful.

Proposition 2.3.7. Let R be of form (2.12) for some Q, B>, γ and µ satisfy-ing the conditions of Proposition 2.3.2 (iv)-(vii) and Proposition 2.3.3 (i)-(iii).Then R is quasi-monotone increasing on K∗.

Proof. Let δ > 0, and define

Rδ(u) = −1

2Q(u, u) +B>(u) + γ −

∫‖ξ‖≥δ∩K

(e−〈u,ξ〉 − 1 + 〈χ(ξ), u〉

)µ(dξ)

= −1

2Q(u, u) + γ +B>(u)−

∫‖ξ‖≥δ∩K

〈χ(ξ), u〉µ(dξ)

−∫‖ξ‖≥δ∩K

(e−〈u,ξ〉 − 1

)µ(dξ).

(2.19)

Take now some u, v ∈ K∗ and x ∈ K such that u v and 〈u, x〉 = 〈v, x〉.Due to Condition (i) of Proposition 2.3.3, we then have 〈Q(v − u,w), x〉 = 0for all w ∈ V . As Q is a bilinear function, this is equivalent to 〈Q(v, w), x〉 =〈Q(u,w), x〉 for all w ∈ V . Inserting w = u and w = v, we obtain by thesymmetry of Q

〈Q(v, v), x〉 = 〈Q(u, u), x〉,whence the map u 7→ −1

2Q(u, u) + γ is quasi-constant. Condition (iii) of

Proposition 2.3.3 directly yields that

u 7→ B>(u)−∫‖ξ‖≥δ∩K

〈χ(ξ), u〉µ(dξ)

is a quasi-monotone increasing linear map on K∗. Finally, the quasi-mono-tonicity of

u 7→∫‖ξ‖≥δ∩K

(1− e−〈u,ξ〉

)µ(dξ)

is a consequence of the monotonicity of the exponential map and supp(µ) ⊆ K.By dominated convergence, we have limδ→0R

δ(u) = R(u) pointwise for eachu ∈ K∗. Hence the quasi-monotonicity carries over to R. Indeed, we have forall δ > 0, 〈Rδ(v)−Rδ(u), x〉 ≥ 0. Thus

〈Rδ(v)−Rδ(u), x〉 → 〈R(v)−R(u), x〉 ≥ 0

as δ → 0, which proves that R is quasi-monotone increasing.


2.4 The Generalized Riccati Equations

In order to prove the existence of affine processes for a given parameter setwhich satisfies the conditions of Proposition 2.3.2 and Proposition 2.3.3, weshall heavily rely on the following existence and uniqueness result for the gen-eralized Riccati equations (2.9) and (2.10), where F and R are given by (2.11)and (2.12). Indeed, in the case of general proper convex cones, this allowsus to prove existence of affine pure jumps processes (see Section 2.5). Inthe particular case of affine processes on symmetric cones, which we study inChapter 3, we obtain, using Theorem 2.4.3 below, existence of affine processesfor any given parameter set (see Section 3.4.2).

For the analysis of the generalized Riccati equations (2.9) and (2.10) weshall use the concept of quasi-monotonicity, as introduced above, several times.Indeed, the proof of Theorem 2.4.3 below is based to a large extent on thefollowing comparison result for ordinary differential equations, which can bededuced from a more general theorem proved by Volkmann [1973].

Theorem 2.4.1. Let U ⊂ V be an open set. Let f : [0, T ) × U → V be acontinuous locally Lipschitz map such that f(t, ·) is quasi-monotone increasingon U for all t ∈ [0, T ). Let 0 < t0 ≤ T and g, h : [0, t0)→ U be differentiablemaps such that g(0) h(0) and

∂tg(t)− f(t, g(t)) ∂th(t)− f(t, h(t)), 0 ≤ t < t0.

Then we have g(t) h(t) for all t ∈ [0, t0).

The following estimate is needed to establish the existence of a globalsolution of (2.10).

Lemma 2.4.2. Let R be of form (2.12) for some Q, B>, γ and µ satisfyingthe conditions of Proposition 2.3.2 (iv)-(vii). Then

R(u) B>(u) + γ + µ(K ∩ ‖ξ‖ > 1).

Proof. We may assume without loss of generality that the truncation functionχ takes the form χ(ξ) = 1‖ξ‖≤1ξ (otherwise we can adjust B>(u) accord-ingly). Then, for all u ∈ K∗, we have

R(u) = −1

2Q(u, u) +B>(u) + γ −

∫K∩‖ξ‖≤1

(e−〈u,ξ〉 − 1 + 〈ξ, u〉

)︸︷︷︸≥0

µ(dξ)

−∫K∩‖ξ‖>1

(e−〈u,ξ〉 − 1

)µ(dξ)

−1

2Q(u, u) +B>(u) + γ + µ(K ∩ ‖ξ‖ > 1)

B>(u) + γ + µ(K ∩ ‖ξ‖ > 1),

2.4. The Generalized Riccati Equations 61

where we use −∫K∩‖ξ‖>1

(e−〈u,ξ〉 − 1

)µ(dξ)

∫K∩‖ξ‖>1 µ(dξ).

Here is our main existence and uniqueness result for the generalized Riccatidifferential equations (2.9)–(2.10).

Theorem 2.4.3. Let F and R be of form (2.11) and (2.12) such that theconditions of Proposition 2.3.2 and 2.3.3 are satisfied. Then, for every u ∈K∗, there exists a unique global R+×K∗-valued solution (φ, ψ) of (2.9)–(2.10).Moreover, φ(t, u) and ψ(t, u) are analytic in (t, u) ∈ R+ × K∗.

Proof. We only have to show that, for every u ∈ K∗, there exists a uniqueglobal K∗-valued solution ψ of (2.10), since φ is then uniquely determined byintegrating (2.9).

Let u ∈ K∗. Since R is analytic on K∗ (see, e.g., Duffie et al. [2003,Lemma A.2]), standard ODE results (see, e.g., Dieudonne [1969, Theorem10.4.5]) yield that there exists a unique local K∗-valued solution ψ(t, u) of(2.10) for t ∈ [0, t∞(u)), where

t∞(u) = limk→∞

inft ≥ 0 | ‖ψ(t, u)‖ ≥ k or ψ(t, u) ∈ ∂K∗ ≤ ∞.

It thus remains to show that t∞(u) =∞. Analyticity of ψ(t, u) and φ(t, u) in(t, u) ∈ R+ × K∗ then follows from Dieudonne [1969, Theorem 10.8.2].

Since R may not be Lipschitz continuous at ∂K∗ (see Remark 2.4.4 below),we first have to regularize it. We thus define

R(u) = −1

2Q(u, u) +B>(u) + γ −

∫K∩‖ξ‖≤1

(e−〈u,ξ〉 − 1 + 〈ξ, u〉

)µ(dξ).

Then R is analytic on V . Hence, for all u ∈ V , there exists a unique localV -valued solution ψ of

∂ψ(t, u)

∂t= R(ψ(t, u)), ψ(0, u) = u,

for all t ∈ [0, t∞(u)) with maximal lifetime

t∞(u) = limk→∞

inft ≥ 0 | ‖ψ(t, u)‖ ≥ k ≤ ∞.

Consider now the normal cone of K∗ at u ∈ ∂K∗, consisting of inward pointingnormal vectors, that is,

NK∗(u) = x ∈ K | 〈u, x〉 = 0, u 6= 0,


and NK∗(0) = K (see, e.g., Hiriart-Urruty and Lemarechal [1993, ExampleIII.5.2.6], except for a change of the sign). The conditions of Proposition 2.3.3thus imply that

〈R(u), x〉 ≥ 0,

for all x ∈ NK∗(u). Since R is clearly Lipschitz continuous, it follows from Wal-

ter [1993, Theorem III.10.XVI] that ψ(t, u) ∈ K∗ for all t < t∞(u) and u ∈ K∗.Let us now define

∂y(t, u)

∂t= B>(y(t, u)) + γ, y(0, u) = u. (2.20)

Then we have by Lemma 2.4.2 for t < t∞(u)

∂ψ(t, u)

∂t− R(ψ(t, u)) =

∂y(t, u)

∂t−B>(y(t, u))− γ ∂y(t, u)

∂t− R(y(t, u)).

Volkmann’s comparison Theorem 2.4.1 thus implies for all x ∈ K

〈ψ(t, u), x〉 ≤ 〈y(t, u), x〉, t ∈ [0, t∞(u)).

As ψ(t, u) lies in K∗ up to its lifetime, the left hand side is nonnegative for allx ∈ K. Moreover, the linear ODE (2.20) admits a global solution. This andthe fact that K is generating yields t∞(u) =∞ for all u ∈ K∗.

Moreover, by Proposition 2.3.7, R is quasi-monotone increasing on K∗.Hence another application of Theorem 2.4.1 yields

0 ψ(t, u) ψ(t, v), t ≥ 0, for all 0 u v.

Therefore and since ψ(t, u) is also analytic in u, Lemma 2.2.1 implies that

ψ(t, u) ∈ K∗ for all (t, u) ∈ R+ × K∗.We now carry this over to ψ(t, u) and assume without loss of generality,

as in the proof of Lemma 2.4.2, that the truncation function χ takes the formχ(ξ) = 1‖ξ‖≤1ξ. Then

R(u)− R(u) = −∫K∩‖ξ‖>1

(e−〈u,ξ〉 − 1

)µ(dξ) 0, u ∈ K∗.

Hence, for u ∈ K∗ and t < t∞(u), we have

∂ψ(t, u)

∂t− R(ψ(t, u)) =

∂ψ(t, u)

∂t−R(ψ(t, u)) ∂ψ(t, u)

∂t− R(ψ(t, u)).

Theorem 2.4.1 thus implies

ψ(t, u) ψ(t, u) ∈ K∗, t ∈ [0, t∞(u)).

2.5. Construction of Affine Pure Jump Processes 63

Hence t∞(u) = limk→∞ inft ≥ 0 | ‖ψ(t, u)‖ ≥ k. Using again Lemma 2.4.2

and the comparison argument with a linear ODE similar as for ψ, we concludethat t∞(u) =∞, as desired.

Remark 2.4.4. Note that quasi-monotonicity just means that R is “inwardpointing” close to the boundary K∗. If R was Lipschitz continuous on K∗, Wal-ter [1993, Theorem III.10.XVI] would imply the invariance of K∗ with respectto (2.10) right away. The difficulty arises from the fact that the map R mightfail to be Lipschitz continuous at ∂K∗ (see the one-dimensional counterexam-ple in Duffie et al. [2003, Example 9.3]), even though it is analytic on theinterior K∗. Here quasi-monotonicity plays the decisive role. It leads to thephenomenon that ψ(t, u) stays away from the boundary ∂K∗ for u ∈ K∗, whichis of crucial importance in our analysis.

2.5 Construction of Affine Pure Jump Pro-

cesses

For affine processes on generating convex proper cones without diffusion com-ponent, that is, Q = 0, the existence question can be handled entirely as in thecase of affine processes on the canonical state space Rm

+ ×Rn−m. By followingthe lines of Duffie et al. [2003, Section 7], we here prove existence of affinepure jump processes for a given parameter set, which satisfies the conditions ofProposition 2.3.2 and Proposition 2.3.3 with the additional assumption Q = 0.

We call a function f : K∗ → R of Levy-Khintchine form on K∗ if

f(u) = 〈b, u〉 −∫K

(e−〈u,ξ〉 − 1)m(dξ),

where b ∈ K and m is a Borel measure supported on K such that∫K

(‖ξ‖ ∧ 1)m(dξ) <∞.

Recall that a distribution on K is infinitely divisible if and only if its Laplacetransform takes the form e−f(u), where f is of the above form. This means –similarly as in the case of R+ – that Levy processes on cones can only be offinite variation.

As in Duffie et al. [2003], let us introduce the sets

C := f + c | f : K∗ → R is of Levy-Khintchine form on K∗ , c ∈ R+,CS := ψ |u 7→ 〈ψ(u), x〉 ∈ C for all x ∈ K .

The following assertion can be obtained easily by mimicking the proofsof Duffie et al. [2003, Proposition 7.2 and Lemma 7.5].


Lemma 2.5.1. We have,

(i) C, CS are convex cones in C(K∗).

(ii) φ ∈ C, ψ ∈ CS imply φ(ψ) ∈ C.

(iii) ψ, ψ1 ∈ CS imply ψ1(ψ) ∈ CS.

(iv) If φk ∈ C converges pointwise to a continuous function φ on K∗, thenφ ∈ C and φ has a continuous extension to K∗. A similar statementholds for sequences in CS.

(v) Let R be of form (2.12) and let Rδ be defined as in (2.19) such that theinvolved parameters satisfy the conditions of Proposition 2.3.3. Then Rδ

converges to R locally uniformly as δ → 0.

Proposition 2.5.2. Let F and R be of form (2.11) and (2.12) such thatthe involved parameters satisfy the conditions of Proposition 2.3.2 and 2.3.3.Then, for all t ≥ 0, the solutions (φ(t, ·), ψ(t, ·)) of (2.9) and (2.10) lie in(C, CS).

Proof. Suppose first that∫K

(‖ξ‖ ∧ 1) 〈x, µ(dξ)〉 <∞, for all x ∈ K. (2.21)

Then equation (2.10) is equivalent to the integral equation

ψ(t, u) = eB>t(u) +

∫ t

0

eB>(t−s)(R(ψ(s, u))ds, (2.22)

where R(u) = R(u) + B>(u) and B> ∈ L(V ) is given by

B>(u) := B>(u)−∫K

〈χ(ξ), u〉µ(dξ).

Here, eB>t(u) is the notation for the semigroup induced by

∂ty(t, u) = B>(y(t, u)), y(0, u) = u.

Hence the variation of constants formula yields (2.22). Due to Proposi-

tion 2.3.3 (iii), B> is a linear drift which is “inward pointing” at the boundary

of K∗. This in turn is equivalent to eB>t being a positive semigroup, that is,

eB>t maps K∗ into K∗. Therefore eB

>t ∈ CS and since R(u) is given by

R(u) = γ −∫K

(e−〈u,ξ〉 − 1)µ(dξ)

2.5. Construction of Affine Pure Jump Processes 65

with µ satisfying (2.21), we also have R ∈ CS.By a classical fixed point argument, the solution ψ(t, u) is the pointwise

limit of the sequence (ψ(k)(t, u))k∈N, for (t, u) ∈ R+ × K∗, obtained by Picarditeration

ψ(0)(t, u) := u,

ψ(k+1)(t, u) := eB>t(u) +

∫ t

0

eB>(t−s)(R(ψ(k)(s, u))ds,

and due to Lemma 2.5.1 (i) and (iii), ψ(k)(t, ·) lies in CS for all k ∈ N. In viewof Lemma 2.5.1 (iv), the limit ψ(t, ·) thus lies in CS as well and there existsa unique continuous extension of ψ on R+ × K∗. Since F ∈ C, we have byLemma 2.5.1 (ii)

φ(t, ·) =

∫ t

0

F (ψ(s, ·))ds ∈ C.

By applying Lemma 2.5.1 (v), the general case is then reduced to the former,since µ1‖ξ‖≥δ clearly satisfies (2.21).

We are now prepared to prove existence of affine processes on generatingconvex proper cones under the additional assumption Q = 0:

Proposition 2.5.3. Suppose that the parameters (Q = 0, b, B, c, γ,m, µ) sat-isfy the conditions of Proposition 2.3.2 and Proposition 2.3.3. Then there ex-ists a unique affine process on K such that (2.1) holds for all (t, u) ∈ R+×K∗,where φ(t, u) and ψ(t, u) are solutions of (2.9) and (2.10) with F and R givenby (2.11) and (2.12).

Proof. By Proposition 2.5.2, (φ(t, ·), ψ(t, ·)) lie in (C, CS). Hence, for all t ∈ R+

and x ∈ K, there exists an infinitely divisible sub-stochastic measure on Kwith Laplace-transform e−φ(t,u)−〈ψ(t,u),x〉. Moreover, the Chapman-Kolmogorovequation holds in view of the flow property of φ and ψ, which implies theassertion.

Chapter 3

Affine Processes on SymmetricCones

We will now considerably strengthen our structural assumptions on the conestate space K and assume that K is a symmetric cone. This assumption allowsus to refine the conditions found in Proposition 2.3.2 and Proposition 2.3.3and enables us to prove existence of affine processes on these particular statespaces. Concerning the refinement of the necessary conditions, we build on ourresults obtained in the setting of positive semidefinite matrices (see Cuchieroet al. [2011a]), while we use a different approach for the treatment of theexistence question.

The setting of symmetric cones, which was introduced in the field of affineprocesses by Grasselli and Tebaldi [2008], covers many important examples,such as the cone Rn

+, the cone of positive semi-definite matrices, and theLorentz cone Λn, defined by Λn := x ∈ Rn |x2

1−∑n

i=2 x2i > 0, x1 > 0. It also

imposes an additional algebraic structure on V and leads to a multiplicationoperation : V × V → V , which endows V with the structure of a so-calledEuclidean Jordan Algebra. The cone K is then exactly the cone of squares inthis algebra, that is, K = x x : x ∈ V .

3.1 Symmetric Cones and Euclidean Jordan

Algebras

We start by explaining the fundamental definitions from the standard refer-ence on symmetric cones and Euclidean Jordan algebras, Faraut and Koranyi[1994]. In order to give some intuition, we always illustrate them by meansof the Euclidean Jordan algebra of r × r real symmetric matrices, which wedenote by Sr. Let us start with the definition of a symmetric cone.

67

68 Chapter 3. Affine Processes on Symmetric Cones

Definition 3.1.1 (Symmetric cone). A convex cone K in an Euclidean space(V, 〈·, ·〉) of dimension n is called symmetric if it is

(i) homogeneous, which means that the automorphism group G(K) = g ∈GL(V ) | gK = K acts transitively, that is, for all x, y ∈ K there existsan invertible linear map g : V → V that leaves K invariant and maps xto y,

(ii) self-dual, that is, K∗ = K.

A symmetric cone K is said to be irreducible if there do not exist non-trivialsubspaces V1, V2 and symmetric cones K1 ⊂ V1, K2 ⊂ V2 such that V is thedirect sum of V1 and V2 and K = K1 +K2.

Example 3.1.2. For illustrative purposes, let us consider the vector spaceSr of dimension n = r(r+1)

2. A scalar product on this space is given by

〈x, y〉 = tr(xy), where tr denotes the usual matrix trace. The correspondingsymmetric cone is the set of positive semidefinite matrices, which we denoteby S+

r . Moreover, we write S++r for the open cone of positive definite matrices.

Clearly, S+r is self-dual with respect to 〈x, y〉 = tr(xy) and its automorphism

group is given by

G(S+r ) =

G ∈ GL(Sr) |Gx = gxg>, g ∈ GL(Rr)

.

Since every matrix x ∈ S++r can be written as x = zz>, where z is an invertible

r × r matrix, it follows that S+r is homogeneous.

Note that a symmetric cone is automatically generating and pointed. Asalready mentioned, symmetric cones are directly related to Euclidean Jordanalgebras, defined as follows:

Definition 3.1.3 (Euclidean Jordan algebra). A real Euclidean space (V, 〈·, ·〉)with a bilinear product : V × V → V : (x, y) 7→ x y and identity element eis called an Euclidean Jordan algebra if

(i) V is a Jordan algebra with product , that is, for all x, y ∈ V

(a) x y = y x, (b) x2 (x y) = x (x2 y),

(ii) and the Jordan product is compatible with the scalar product in the sensethat

〈x y, z〉 = 〈y, x z〉.

An Euclidean Jordan algebra is said to be simple if it does not contain anynon-trivial ideal.

3.1. Symmetric Cones and Euclidean Jordan Algebras 69

Remark 3.1.4. Note that the Jordan product is commutative by (a), butin general not associative. Thus (b) is a genuine axiom. We have used x2

to denote the Jordan product x x. This should not cause confusion, evenwhen we use the same notation to denote powers of scalars and matrices. Byinduction it is seen that V is a power associative algebra, that is, xm xn =xn xm = xm+n, for all m, n ≥ 1.

Example 3.1.5. By defining the following product on the vector space of r×rreal symmetric matrices

x y =1

2(xy + yx),

it is easily verified that Sr is a Jordan algebra. Here xy denotes the usual ma-trix multiplication. Note also that the scalar product satisfies 〈x, y〉 = tr(xy) =tr(yx) = tr(x y).

For an element x ∈ V we introduce the left-product operator, denoted byL and defined by

L(x)y = x y.

Moreover, P denotes the so-called quadratic representation of V , given by

P (x) = 2L(x)2 − L(x2). (3.1)

For both operators we have L = L> and P = P>. In the case of Sr, thequadratic representation is given by P (x)y = xyx.

The one-to-one correspondence between Euclidean Jordan algebras andsymmetric cones is established in Faraut and Koranyi [1994, Theorem III.3.1,III.4.4 and III.4.5] and can be rephrased as follows:

Theorem 3.1.6. Let K be a symmetric cone in V . Then there exists a Jordanproduct on V such that (V, ) is an Euclidean Jordan algebra, and

K = x2 : x ∈ V .

The symmetric cone is irreducible if and only if the associated Euclidean Jor-dan algebra is simple.

Example 3.1.7. In the case of S+r , the above theorem can easily be verified,

sinceS+r = x x = x2 |x ∈ Sr.

Note that the Jordan product x x = x2 equals the matrix product xx = x2 inthis case. This is positive semidefinite, since the eigenvalues of x2 are givenby the squared eigenvalues of x.


For further definitions and results on Jordan algebras and symmetric cones,which we shall use in the subsequent sections, we refer to Faraut and Koranyi[1994] and Appendix A. There we give an overview of the most importantresults of the theory of Jordan algebras and illustrate them by means of real-valued symmetric matrices. Our terminology and notation follows the bookof Faraut and Koranyi [1994].

3.2 Refinement of the Necessary Conditions

Throughout this section we suppose that X is an affine process on some ir-reducible symmetric cone K and V denotes the associated simple EuclideanJordan algebra of dimension n and rank r, equipped with the natural scalarproduct

〈·, ·〉 : V × V → R, 〈x, y〉 := tr(x y).

For the notion of the trace, denoted by tr, we refer to Appendix A.1.1. We shallalso use the Peirce invariant d corresponding to the dimension of Vij, i < j,as defined in (A.7). In our case of a simple Euclidean Jordan algebra, we thenhave n = r + d

2r(r − 1). For the precise definitions we refer to Appendix A.

Using the additional Euclidean Jordan algebra structure, we aim to im-prove the parameter conditions derived in Proposition 2.3.2 and Proposi-tion 2.3.3 such that we finally obtain conditions which guarantee existenceof affine processes on symmetric cones. The focus lies in particular on thebilinear form Q corresponding to the linear diffusion part, on the linear jumpcoefficient µ and on the constant drift part b.

3.2.1 Representation of the Diffusion Part

The following proposition establishes a direct relation between the bilinearform Q satisfying the condition of Proposition 2.3.3 and the quadratic repre-sentation of V . For its proof we use the Peirce and spectral decomposition ofan Euclidean Jordan algebra, as introduced in Appendix A.1.2.

Proposition 3.2.1. Let V be a simple Euclidean Jordan algebra of rank rand let Q : V ×V → V be a symmetric bilinear function with Q(v, v) ∈ K forall v ∈ V . Then Condition (i) of Proposition 2.3.3, that is,

〈x,Q(u, v)〉 = 0 for all v ∈ V and u, x ∈ K with 〈u, x〉 = 0, (3.2)

is satisfied if and only if

Q(u, u) = 4P (u)α.

3.2. Refinement of the Necessary Conditions 71

Here, P (u) is the quadratic representation of the Jordan algebra V , definedin (3.1) and α ∈ K is determined by 4α = Q(e, e).

Remark 3.2.2. (i) In the case of V = Sr, the above proposition meansthat Condition (i) of Proposition 2.3.3 is equivalent to Q(u, u) = 4uαu,where α ∈ S+

r .

(ii) The above assertion has also been obtained in Grasselli and Tebaldi[2008], but has been proved differently therein.

Proof. We first assume that (3.2) is satisfied. Let u ∈ V be fixed. Then thereexists a Jordan frame p1, . . . , pr (see Appendix A.1.2 for the precise definition)such that its spectral decomposition is given by u =

∑ri=1 λipi.

As p1 + · · ·+ pr = e, we can write Q(e, e) as

Q(e, e) = Q(p1, p1) +Q(p2, p2) + · · ·+Q(pr, pr) +∑i<j

2Q(pi, pj). (3.3)

We now show that (3.3) is precisely the Peirce decomposition ofQ(e, e) with re-spect to the Jordan frame p1, . . . , pr. More precisely, we show that Q(pj, pj) ∈Vjj and Q(pi, pj) ∈ Vij for each i, j ∈ 1, . . . , r.

Let i 6= j, then clearly 〈pi, pj〉 = 0. From (3.2) we deduce that

〈pi, Q(pj, pj)〉 = 0. (3.4)

But Q(pj, pj) ∈ K such that Q(pj, pj) pi = 0 by Lemma A.2.1. Keeping jfixed, we can subtract these equalities from Q(pj, pj) e = Q(pj, pj), runningthrough all i 6= j, and we arrive at Q(pj, pj) pj = Q(pj, pj). This shows thatQ(pj, pj) ∈ V (pj, 1) = Vjj for all j ∈ 1, . . . , r.

Let now i, j, k be arbitrary in 1, . . . , r, but all distinct. Using again (3.2),we see that 〈pi, Q(pk +pj, pk +pj)〉 = 0, and from Lemma A.2.1 it follows thatQ(pk + pj, pk + pj) pi = 0. Thus

Q(pk, pj)pi =1

2(Q(pk + pj, pk + pj) pi −Q(pk, pk) pi −Q(pj, pj) pi) = 0

for any distinct i, j, k ∈ 1, . . . , r. Keeping now k and j fixed, we can subtractthe equalities Q(pk, pj) pi = 0 from the equality Q(pk, pj) e = Q(pk, pj),running through all i distinct from both j and k, and obtain Q(pk, pj) (pk +pj) = Q(pk, pj). For symmetry reasons we must haveQ(pk, pj)pk = Q(pk, pj)pj and we thus conclude that

Q(pk, pj) pk = Q(pk, pj) pj =1

2Q(pk, pj).


Equivalently, Q(pk, pj) ∈ V (pk, 1/2) ∩ V (pj, 1/2) = Vkj, and we have shownthat (3.3) is the Peirce decomposition of Q(e, e) with respect to the Jordanframe p1, . . . , pr.

Define 4α := Q(e, e). As the projection onto Vii is given by the quadraticrepresentation P (pi) and the projection onto Vij by 4L(pi)L(pj), we can writeQ(pj, pj) = 4P (pj)α and 2Q(pi, pj) = 16L(pi)L(pj)α. Therefore,

Q(u, u) = λ21Q(p1, p1) + · · ·+ λ2

rQ(pr, pr) +∑i<j

2λiλjQ(pi, pj)

= 4

(λ2

1P (p1)α + · · ·+ λ2rP (pr)α +

∑i<j

λiλj4L(pi)L(pj)α

)

= 4P

(r∑i=1

λipi

)α

= 4P (u)α,

and we have shown the first implication.Concerning the other direction, let Q be given by Q(u, u) = 4P (u)α for

some α ∈ K. Using polarization, we then get

Q(u, v) = 2 (P (u+ v)− P (u)− P (v))α.

Take now some x, u ∈ K such that 〈x, u〉 = 0. By Lemma A.2.1 (ii), we haveu x = 0 and consequently

〈x, u2〉 = 〈x u, u〉 = 0,

which in turn implies u2x = 0. The definition of the quadratic representationthus yields

〈P (u)α, x〉 = 〈α, P (u)x〉 = 〈α, 2u (u x)− u2 x〉 = 0.

Similarly, we have 〈P (u+v)α, x〉 = 〈P (v)α, x〉, which proves the assertion.

3.2.2 Linear Jump Coefficient

In this section, we show that the linear jump coefficient µ satisfying Condi-tion (vii) of Proposition 2.3.2 and Condition (ii) of Proposition 2.3.3 neces-sarily integrates (‖ξ‖ ∧ 1) if r > 1 and d > 0. The proof is based on anidea of Mayerhofer [2011], who showed the corresponding result for positivesemidefinite matrices.


Proposition 3.2.3. Let V be a simple Euclidean Jordan algebra with rankr > 1 and Peirce invariant d > 0. Suppose that µ is a K-valued σ-finite Borelmeasure on K satisfying µ(0) = 0 and∫

K

(‖ξ‖2 ∧ 1)〈x, µ(dξ)〉 <∞, for all x ∈ K.

Then Condition (ii) of Proposition 2.3.3, that is,∫K

〈χ(ξ), u〉〈x, µ(dξ)〉 <∞ for all u, x ∈ K with 〈u, x〉 = 0, (3.5)

implies ∫K

(‖ξ‖ ∧ 1)〈x, µ(dξ)〉 <∞, for all x ∈ K.

Remark 3.2.4. It follows from the above proposition that only in the caseof R+ and the two-dimensional Lorentz cone, jumps of infinite variation arepossible. In all other cases we could now set the truncation function χ tobe 0 and adjust the linear drift accordingly. However, in order to cover allirreducible cones, we shall keep the truncation function in the sequel.

Proof. Let p1, . . . , pr be a fixed Jordan frame of V . Corresponding to thePeirce decomposition (A.7), we can write for every z ∈ V

z =r∑i=1

zipi +∑i<j

zij,

where zi ∈ R and zij ∈ Vij. For the K-valued measure µ, this means that,for every i ∈ 1, . . . , r, µi denotes a positive measure and, for i 6= j, µij is aVij-valued measure. Consider now, for some i 6= j, elements of the form

x = pi + pj + w, u = pi + pj − w,

with w ∈ Vij such that ‖w‖2 = 2, which implies that x, u ∈ ∂K and 〈u, x〉 = 0.Assume without loss of generality that χ(ξ) = 1‖ξ‖≤1ξ. Since for every y ∈ K,〈y, µ(·)〉 is a positive measure supported on K, we have by (3.5)

0 ≤∫‖ξ‖≤1

〈ξ, u〉〈x, µ(dξ)〉 <∞,

0 ≤∫‖ξ‖≤1

〈ξ, x〉〈u, µ(dξ)〉 <∞.


Thus there exists a positive constant C such that for all δ > 0

0 ≤∫δ≤‖ξ‖≤1

〈ξ, u〉〈x, µ(dξ)〉 < C, (3.6)

0 ≤∫δ≤‖ξ‖≤1

〈ξ, x〉〈u, µ(dξ)〉 < C. (3.7)

Summing up (3.6) and (3.7) and using the orthogonality of the Peirce decom-position, then yields

0 ≤∫δ≤‖ξ‖≤1

(ξiµi(dξ)−

1

2〈ξij, w〉〈w, µij(dξ)〉+ ξjµj(dξ)〉

)+

∫δ≤‖ξ‖≤1

(ξiµj(dξ)−

1

2〈ξij, w〉〈w, µij(dξ)〉+ ξjµi(dξ)

)< 2C. (3.8)

Since µ is a K-valued measure on K, we have by Faraut and Koranyi [1994,Exercise IV.7 (b)] and the assumption ‖w‖2 = 2

1

2〈ξij, w〉〈w, µij〉 ≤ ‖ξij‖‖µij‖ ≤ 2

√ξiξjµiµj,

which implies that both integrals in (3.8) are nonnegative. We can thereforeconclude that both of them are finite:

0 ≤∫δ≤‖ξ‖≤1

(ξiµi(dξ)−

1

2〈ξij, w〉〈w, µij(dξ)〉+ ξjµj(dξ)〉

)< 2C, (3.9)

0 ≤∫δ≤‖ξ‖≤1

(ξiµj(dξ)−

1

2〈ξij, w〉〈w, µij(dξ)〉+ ξjµi(dξ)

)< 2C. (3.10)

Moreover, as 〈pi, pj〉 = 0 for i 6= j, we have as a direct consequence of (3.5)

0 ≤∫‖ξ‖≤1

〈ξ, pi〉〈pj, µ(dξ)〉 =

∫‖ξ‖≤1

ξiµj(dξ)〉 <∞, i 6= j. (3.11)

As above, there thus exists a positive constant C1 such that for all δ > 0∫δ≤‖ξ‖≤1

ξiµj(dξ) < C1, i 6= j. (3.12)

Subtracting (3.10) from (3.12), then yields for all δ > 0

−2C <

∫δ≤‖ξ‖≤1

1

2〈ξij, w〉〈w, µij(dξ)〉 < 2C1.


By (3.9), we therefore have for all δ > 0

0 ≤∫δ≤‖ξ‖≤1

(ξiµi(dξ) + ξjµj(dξ)) < 2(C + C1).

Together with (3.11), this implies for all i ∈ 1, . . . , r

0 ≤∫‖ξ‖≤1

ξiµi(dξ) <∞.

This then yields ∫K

(‖ξ‖ ∧ 1)〈x, µ(dξ)〉 <∞, for all x ∈ K,

and proves the assertion.

3.2.3 The Special Role of the Constant Drift Part

This section is devoted to show that the constant drift term b, as defined inProposition 2.3.2 (i), of any affine process X on an irreducible symmetric coneK necessarily satisfies

b d(r − 1)α,

where α is defined in Proposition 3.2.1. Recall that d denotes the Peirceinvariant and r the rank of V .

Before we actually prove this result, let us introduce some notation. Weshall consider the tensor product V ⊗ V ∗, which we identify via the canonicalisomorphism with the vector space of linear maps on V denoted by L(V ).Moreover, for an element A ∈ L(V ), we denote its trace by Tr(A).1 Observethat Tr(A(u⊗ u)) = 〈u,Au〉. Indeed, by choosing a basis eβ of V , we have

Tr(A(u⊗ u)) =∑β

〈A>eβ, (u⊗ u)eβ〉

=∑β

〈A>eβ, 〈u, eβ〉u〉

=∑β

〈u, eβ〉〈A>eβ, u〉

= 〈u,Au〉.1In order to distinguish between elements of V and linear maps on V , we use the no-

tations Tr(A) and Det(A) for A ∈ L(V ) and tr(x) and det(x) for elements in V (compareRemark A.1.1).


Let now A : K → S+(V ) ⊂ L(V ) be the linear part of the diffusion charac-teristic, as introduced in (2.15). Recall that the symmetric bilinear functionQ was defined via (2.18), that is,

Tr(A(x)(u⊗ u)) = 〈u,A(x)u〉 = 〈x,Q(u, u)〉.

As shown in Proposition 3.2.1, we have Q(u, u) = 4P (u)α for some α ∈ K.Hence

Tr(A(x)(u⊗ u)) = 〈u,A(x)u〉 = 4〈x, P (u)α〉. (3.13)

We can now define a second order differential operator for this expressionand introduce the following integro-differential operator for (complex-valued)C2b (K)-functions.

Af(x) =1

2Tr

(A(x)

(∂

∂x⊗ ∂

∂x

))f |x + 〈b+B(x),∇f(x)〉

− (c+ 〈γ, x〉)f(x) +

∫K

(f(x+ ξ)− f(x))m(dξ)

+

∫K

(f(x+ ξ)− f(x)− 〈χ(ξ),∇f(x)〉) 〈x, µ(dξ)〉,

(3.14)

where A(x) satisfies (3.13). The other parameters are specified in Proposi-tion 2.3.2 and are supposed to satisfy the conditions of Proposition 2.3.3 andPropsition 3.2.3. Note that for the family of functions e−〈u,x〉 |u ∈ K thisexpression corresponds to the pointwise t-derivative of Pte

−〈u,x〉 at t = 0. Thisis simply a consequence of the form of F and R, since

limt↓0

(Pte−〈u,x〉 − e−〈u,x〉

)t

= (−F (u)− 〈R(u), x〉)e−〈u,x〉 = Ae−〈u,x〉

for every x ∈ K. In view of Definition 1.4.1, A corresponds to the extendedgenerator of the affine process X.

The following lemma is proved by means of the Levy–Khintchine formulaon R+ and is related to the positive maximum principle for the operator A.

Lemma 3.2.5. Let X be an affine process on K with constant drift parameterb and linear diffusion part Q, as defined in Proposition 2.3.2 (i) and (iv).Moreover, suppose that Q satisfies Q(u, u) = 4P (u)α for some α ∈ K. Then,for any y ∈ ∂K, we have

〈b,∇ det(y)〉+ 2

⟨y, P

(∂

∂x

)α

⟩det |y

= 〈b,∇ det(y)〉+1

2Tr

(A(y)

(∂

∂x⊗ ∂

∂x

))det |y ≥ 0.

(3.15)


Here, det(y) denotes the determinant of an element y ∈ V , as defined inAppendix A.1.1, and A(x) is the linear part of the diffusion characteristic,which satisfies

Tr(A(x)(u⊗ u)) = 〈u,A(x)u〉 = 〈x,Q(u, u)〉 = 4〈x, P (u)α〉 (3.16)

for all u, x ∈ K.

Proof. Let y ∈ ∂K and let f ∈ C∞c (V ) be a function with f ≥ 0 on K andf(x) = det(x) for all x in a neighborhood of y. Then, for any v ∈ R+, thefunction x 7→ (e−vf(x)− 1) lies in C∞c (V ), hence in particular in Sn, the spaceof rapidly decreasing C∞-functions on V . As the Fourier transform is a linearisomorphism on Sn, we can write

e−vf(y) − 1 =

∫V

ei〈q,y〉g(q)dq

for some g ∈ Sn. As a consequence of Theorem 1.5.4, Remark 2.1.2 andRemark 2.2.4, we obtain by dominated convergence

limt↓0

Pt(e−vf(y) − 1)

t= ∂t|t=0Pt(e

−vf(y) − 1) =

∫V

∂t|t=0Ptei〈q,y〉g(q)dq

=

∫V

(F(i q) + 〈R(i q), y〉) ei〈q,y〉g(q)dq

=

∫V

(−F (− i q)− 〈R(− i q), y〉) ei〈q,y〉g(q)dq

=

∫V

Aei〈q,y〉g(q)dq = A(e−vf(y) − 1),

whereA is defined in (3.14) and thus satisfies (−F (− i q)−〈R(− i q), x〉)ei〈q,x〉 =Aei〈q,x〉. Hence the limit

A(e−vf(y) − 1) = limt↓0

1

t

∫K

(e−vf(ξ) − 1)pt(y, dξ)

= limt↓0

1

t

∫R+

(e−vz − 1)pft (y, dz),(3.17)

exists for any v ∈ R+, where pft (y, dz) = f∗pt(y, dz) is the pushforward ofpt(y, ·) under f , which is a probability measure supported on R+. Using thesame arguments as in the proof of Proposition 2.3.2, we see that, for every fixedt > 0, the right hand side of (3.17) is the logarithm of the Laplace transformof a compound Poisson distribution supported on R+ with intensity 1/t andcompounding distribution pft (y, dz). The pointwise convergence of (3.17) for


t → 0 to some function being continuous at 0 implies weak convergence ofthe compound Poisson distributions to some infinitely divisible probabilitydistribution supported on R+. Its Laplace transform is then given as theexponential of the left hand side of (3.17).

Using now f(y) = 0 and the form of A given by (3.14), we have

v 7→ A(e−vf(y) − 1)

= 2v2 〈y, P (∇f(y))α〉 − 2v

⟨y, P

(∂

∂x

)α

⟩f |y

− v〈b+B(y),∇f(y)〉+

∫K

(e−vf(y+ξ) − 1

)m(dξ)

+

∫K

(e−vf(y+ξ) − 1 + v〈χ(ξ),∇f(y)〉

)〈y, µ(dξ)〉.

(3.18)

Note now that 〈∇ det(y), y〉 = 0 such that the admissibility Conditions (ii)and (iii) of Proposition 2.3.3 imply2∫

K

〈χ(ξ),∇ det(y)〉〈y, µ(dξ)〉 <∞

and

B(y) = 〈y,B>(∇ det(y))〉 −∫K

〈χ(ξ),∇ det(y)〉〈y, µ(dξ)〉 ≥ 0.

By the Levy–Khintchine formula on R+, 〈y, P (∇f(y))α〉 has to vanish, whichis the case due to Proposition 3.2.1 and the fact that 〈∇ det(y), y〉 = 0. More-over, the coefficient of v in (3.18) has to be non-positive, that is,

2

⟨y, P

(∂

∂x

)α

⟩f |y + 〈b+B(y),∇f(y)〉 −

∫K

〈χ(ξ),∇f(y)〉〈y, µ(dξ)〉 ≥ 0.

Observing that y 7→ B(y) is a polynomial of degree r, being positive for everyy ∈ ∂K, and that the polynomial

y 7→ 〈b,∇ det(y)〉+ 2

⟨y, P

(∂

∂x

)α

⟩det |y

is of degree r − 1, we obtain Condition (3.15).

2By Proposition 3.2.3, we have∫K‖χ(ξ)‖〈y, µ(dξ)〉 <∞ if r > 1 and d > 0. This means

that the above argument using 〈∇ det(y), y〉 = 0 is only relevant in the two-dimensionalLorentz cone. Observe that for K = R+, y = 0 anyway.


Proposition 3.2.6. Let X be an affine process on K with constant drift pa-rameter b ∈ K and diffusion parameter α ∈ K, which defines Q(u, u) throughQ(u, u) = 4P (u)α. Then

b d(r − 1)α,

where d denotes the Peirce invariant and r the rank of V .

Remark 3.2.7. Since d = 1 for V = Sr, the above conditions reads in thiscase as

b (r − 1)α,

where α is specified in Remark 3.2.2 (i).

Proof. From Lemma 3.2.5 we have the necessary condition

〈b,∇ det(y)〉+1

2Tr

(A(y)

(∂

∂x⊗ ∂

∂x

))det |y ≥ 0

for any y ∈ ∂K. For x ∈ K we can calculate the left hand side. Since∇ det(x) = det(x)x−1 and d

dt(x−1 +tu)|t=0 = −P (x−1)u (Proposition A.2.2 (v)

and (iii)), we have

〈b,∇ det(x)〉+1

2Tr

(A(x)

(∂

∂x⊗ ∂

∂x

))det |x

= det(x)

(⟨x−1, b

⟩+

1

2Tr(A(x)

(x−1 ⊗ x−1

))− 1

2Tr(A(x)P

(x−1)))

.

Using (3.16), Proposition A.2.2 (i) and Lemma 3.2.8 below, we thus obtain

det(x)

(⟨x−1, b

⟩+ 2

⟨x, P

(x−1)α⟩− 1

2Tr(A(x)P

(x−1)))

= det(x)(⟨x−1, b

⟩+ 2

⟨x−1, α

⟩− 2

n

r

⟨x−1, α

⟩)= det(x)

(⟨x−1, b

⟩− d(r − 1)

⟨x−1, α

⟩)= det(x)

⟨x−1, b− d(r − 1)α

⟩.

As det(y)y−1 is also well-defined on ∂K, Condition (3.15) implies

b d(r − 1)α.

The following lemma is needed in the proof of the above proposition andallows us to express Tr (A(x)P (x−1)) in terms of α.


Lemma 3.2.8. Let V be a simple Euclidean Jordan algebra of rank r andwith scalar product 〈x, y〉 = tr(x y) and let A(x) be defined by (3.16). Then

Tr(A(x)P

(x−1))

= 4n

r

⟨x−1, α

⟩(3.19)

for any invertible x ∈ V .

Proof. Let p1, . . . , pr be a Jordan frame of V . Then the spectral decompositionof an arbitrary element x is given by x =

∑ri=1 λipi, and P (x−1) can be written

as

P(x−1)

=r∑i=1

λ−2i P (pi) +

∑i<j

4λ−1i λ−1

j L(pi)L(pj). (3.20)

Let now eβ be an orthonormal basis of V , where the basis elements arechosen to lie in the subspaces corresponding to the Peirce decomposition, asdescribed in Section A.1.2. More precisely, for each i ≤ r, we choose onebasis element in Vii, which is in fact pi, and for each i < j, we choose d basiselements in Vij, since the dimension of Vij is d.

By the definition of the trace Tr, we have

Tr(A(x)P (x−1)

)=∑β

⟨A(x)eβ, P (x−1)eβ

⟩.

In order to evaluate P (x−1)eβ, we shall use

Vii = x ∈ V |L(pk)x = δikx,

Vij =

x ∈ V |L(pk)x =

1

2(δik + δjk)x

,

as derived in the proof of Faraut and Koranyi [1994, Theorem IV.2.1]. Thisimplies for eβ ∈ Vij, i ≤ j,

L(pk)eβ =1

2(δikeβ + δjkeβ), (3.21)

and hence for k < l,

L(pl)L(pk)eβ =

14eβ, if eβ ∈ Vkl,0, otherwise,

P (pk)eβ =

eβ, if eβ ∈ Vkk,0, otherwise.

Note that this is obvious, since P (pk) and 4L(pl)L(pk) are the orthogonalprojections on Vkk and Vkl respectively (see Section A.1.2).


Let now eβ ∈ Vij, i ≤ j be fixed. Then, using (3.20), the linearity of Aand (3.16), we obtain⟨

A(x)eβ, P(x−1)eβ⟩

=

⟨r∑

k=1

λkA(pk)eβ, λ−1i λ−1

j eβ

⟩

=r∑

k=1

λkλ−1i λ−1

j 4〈P (eβ)pk, α〉

= 2(⟨λ−1i pi, α

⟩+⟨λ−1j pj, α

⟩). (3.22)

Here, the last equality follows from

P (eβ)pk =1

2(δikpj + δjkpi), (3.23)

for eβ ∈ Vij, i ≤ j. For eβ ∈ Vii, (3.23) is simply a consequence of (3.21) andfor eβ ∈ Vij, i < j, we have by Faraut and Koranyi [1994, Proposition IV.1.4(i)]

P (eβ)pk = e2β (δike+ δjke− pk) =

1

2(pi + pj) (δike+ δjke− pk),

which then leads to (3.23). By summing over all basis elements, we deducefrom (3.22)

Tr(A(x)P

(x−1))

=r∑i=1

(1 +

d

2(r − 1)

)4⟨λ−1i pi, α

⟩=n

r4〈x−1, α〉,

where the last equality follows from the fact that n = r + d2r(r − 1).

Remark 3.2.9. If V is an associative Euclidean Jordan algebra, then (3.19)can be derived in a simpler way. Defining a Jordan product by

x y =1

2(xy + yx),

implies P (x)y = xyx and P (x, y)z = 12(xzy + yzx) (see Faraut and Koranyi

[1994, page 32]). Using now (3.16), yields

Tr(A(x)P (x−1)

)=∑β

⟨A(x)eβ, P (x−1)eβ

⟩=∑β

4⟨x, P

(P (x−1)eβ, eβ

)α⟩

=∑β

2⟨x, x−1eβx

−1αeβ + eβαx−1eβx

−1⟩

=∑β

4⟨e2β, x

−1 α⟩

=n

r4⟨x−1, α

⟩,


since∑

β e2β = n

re (see Faraut and Koranyi [1994, e.g., Proof of Proposition

VI.4.1]).In general, Lemma 3.2.8 cannot be proved by these simpler arguments,

since the Lorentz cone and the 3 × 3 Hermitian matrices over the octonionsare examples of non-associative simple Euclidean Jordan algebras.

3.3 Discussion of the Admissibility Conditions

In the following definition we summarize the above derived necessary param-eter restrictions for affine processes on symmetric cones.

Definition 3.3.1 (Admissible parameters). An admissible parameter set(α, b, B, c, γ,m, µ) (associated with a truncation function χ) for an affine pro-cess on an irreducible symmetric cone K consists of

• a linear diffusion coefficient

α ∈ K, (3.24)

• a constant drift termb d(r − 1)α, (3.25)

• a constant killing rate term

c ∈ R+, (3.26)

• a linear killing rate coefficient

γ ∈ K, (3.27)

• a constant jump term: a Borel measure m on K satisfying

m(0) = 0 and

∫K

(‖ξ‖ ∧ 1)m(dξ) <∞, (3.28)

• a linear jump coefficient: a K-valued σ-finite Borel measure µ on Kwith µ(0) = 0 such that the kernel

M(x, dξ) = 〈x, µ(dξ)〉 (3.29)

satisfies ∫K

(‖ξ‖2 ∧ 1)M(x, dξ) <∞, for all x ∈ K,

and∫K

〈χ(ξ), u〉M(x, dξ) <∞ for all x, u ∈ K with 〈x, u〉 = 0, (3.30)

3.3. Discussion of the Admissibility Conditions 83

• a linear drift coefficient: a linear map B> : V → V such that

〈x,B>(u)〉−∫K

〈χ(ξ), u〉M(x, dξ) ≥ 0 for all x, u ∈ K with 〈x, u〉 = 0.

(3.31)

Remark 3.3.2. Note that (3.30) and (3.31) can be considerably simplified ifdimV > 2. In this case r > 1 and d > 0 such that Proposition 3.2.3 allows toset χ = 0.

Using the above definition, let us collect the results that we derived so far.

Theorem 3.3.3. Let X be an affine process on an irreducible symmetric coneK. Then X is regular and has the Feller property. Moreover, φ and ψ satisfythe generalized Riccati equations (2.9) and (2.10) and there exists an admis-sible parameter set (α, b, B, c, γ,m, µ) such that the functions F and R are ofthe following form

R(u) = −2P (u)α +B>(u) + γ −∫K

(e−〈ξ,u〉 − 1 + 〈χ(ξ), u〉

)µ(dξ), (3.32)

F (u) = 〈b, u〉+ c−∫K

(e−〈ξ,u〉 − 1

)m(dξ). (3.33)

Proof. The result is a consequence of Theorem 2.2.2, Theorem 2.2.5, Propo-sition 2.3.2, Proposition 2.3.3, Proposition 3.2.1 and Proposition 3.2.6.

In order to give some intuition on the above defined admissibility condi-tions, we discuss and highlight some properties of the admissible parameterset (α, b, B, c, γ,m, µ). In particular, we shall compare them with the well-known admissibility conditions for the canonical state space Rm

+ × Rn−m andthe cone Rm

+ (see Duffie et al. [2003, Definition 2.6]). Note that the latter isa reducible symmetric cone, whose associated Euclidean Jordan algebra Rm

is of rank 1. We also exemplify the admissibility conditions by means of thecone of r × r positive semidefinite matrices.

3.3.1 Diffusion

As we have already seen in Proposition 2.3.2, the diffusion term does notadmit a constant part if the state space is a general proper convex cone. Thisis one difference to the mixed state space Rn×Rm

+ , where a constant diffusionpart is possible.

Condition (i) of Proposition 2.3.3, that is,

〈x,Q(u, v)〉 = 0 for all v ∈ V and u, x ∈ K with 〈u, x〉 = 0,


is simply a consequence of parallel diffusion behavior along the boundary ofthe state space. In the case of symmetric cones, this then translates to

〈u,A(x)u〉 = 〈x,Q(u, u)〉 = 4〈x, P (u)α〉, α ∈ K, u ∈ V. (3.34)

This property is also in line with the admissibility conditions on the re-ducible symmetric cone Rm

+ . In this case the diffusion part A(x) is of the formA(x) =

∑mi=1 4αieixi. Here, αi ∈ R+ and ei denotes the matrix, where the

(ii)th entry is 1 and all others are 0. As the quadratic representation of Rm isgiven by

P (x)y = (x21y1, . . . , x

2mym)>, x, y ∈ Rm,

relation (3.34) also holds for the cone Rm+ . In the case of positive semidefinite

matrices, the above simplifies to 〈u,A(x)u〉 = 4〈x, uαu〉.

3.3.2 Drift

The remarkable drift Condition (3.25) can be explained by the fact that theboundary of a symmetric cone is curved and kinked, implying this trade-offbetween diffusion coefficient α and b. Indeed, this condition is a consequenceof the positive maximum principle for the (extended) generator A, definedin (3.14) (see Lemma 3.2.5).

Note that, for the rank 1 Jordan algebra R and the symmetric cone R+,the drift condition simply reduces to the non-negativity of b. Therefore weonly have b ∈ Rm

+ if K = R+m.

Let us now consider the cone of positive semidefinite r × r matrices withr > 1. In this case the Peirce invariant equals 1, whence b (r − 1)α. In theliterature, the linear drift part B(x) is usually assumed to be of the form

B(x) = Hx+ xH> (3.35)

for some r × r matrix H. Then we have

〈B(x), u〉 = 〈Hx+ xH>, u〉 = 0 for all x, u ∈ S+r with 〈x, u〉 = 0, (3.36)

such that (3.31) is satisfied. Note that, due to Proposition 3.2.3, the truncationfunction χ can set to be 0. Another possible specifcation of B(x), such thatCondition (3.31) still holds true, is

B(x) = Hx+ xH> + Γ(x), (3.37)

where Γ : Sr → Sr is a linear map satisfying Γ(S+r ) ⊆ S+

r . Here is a simpleexample where B(x) is of the form (3.37) but not of the usual form (3.35): letr = 2 and

B(x) =

(x22 x12

x12 x11

).

3.3. Discussion of the Admissibility Conditions 85

It can be easily checked that (3.31) is satisfied, while B(x) cannot be broughtinto the form (3.35).

3.3.3 Killing

A necessary condition for an affine process on any convex proper cone tobe conservative is c = 0 and γ = 0. In the case of symmetric cones, itcan be proved as in Mayerhofer, Muhle-Karbe, and Smirnov [2011] that X isconservative if and only if c = 0 and ψ(t, 0) ≡ 0 is the only K-valued localsolution of (2.10) for u = 0. The latter condition clearly requires that γ = 0.

A sufficient condition for X to be conservative is c = 0, γ = 0 and

∫K∩‖ξ‖≥1

‖ξ‖M(x, dξ) <∞, for all x ∈ K.

Indeed, it can be shown similarly as in Duffie et al. [2003, Section 9] that thelatter property implies Lipschitz continuity of R(u) on K.

3.3.4 Jumps

For general convex proper cones, Condition (3.28) means that jumps describedby m, which can for instance appear at x = 0, should be of finite variationentering the cone K, since infinite variation transversal to the boundary couldmake the process leave the state space. Similarly, Condition (3.30) assertsfinite variation for the inward pointing direction. However, due to the geome-try of irreducible symmetric cones in dimensions n > 2, such a behavior is nolonger possible and all jumps are in fact of finite total variation (see Propo-sition 3.2.3). Note however that in the case of R+ and the two-dimensionalLorentz cone, the linear jump part can have infinite total variation (see [Duffieet al., 2003, Equation (2.11)]).

Let us also remark that for r > 1 and d > 0, affine diffusion processescannot be approximated (in law) by pure jump processes, since this wouldyield a contradiction to Condition (3.25). However, such an approximation ispossible for the canonical state space, since the rank of the Euclidean Jordanalgebra Rn is 1. This is explicitly exploited in the existence proof for affineprocesses on Rm

+ × Rn−m (see Duffie et al. [2003, Section 7]).


3.4 Construction of Affine Processes on Sym-

metric Cones

Throughout this section we use the same setting as in the previous one, thatis, we suppose that V is a simple Euclidean Jordan algebra of dimension nand rank r and K is the associated irreducible symmetric cone. As before,we assume that the scalar product on V is defined by 〈x, y〉 = tr(x y) andthe Peirce invariant d corresponds to the dimension of Vij, i < j, as defined inAppendix (A.7). Again we refer to Faraut and Koranyi [1994] and Appendix Afor results on Euclidean Jordan algebras.

3.4.1 Construction of Affine Diffusion Processes

The aim of this section is to establish existence of affine diffusion processes forthe following admissible parameter set (α, δα, 0, 0, 0, 0, 0) with δ ≥ d(r − 1).To this end we consider the Riccati equations for φ and ψ associated to theseparameters and show that, for every (t, x) ∈ R+ × K, e−φ(t,u)−〈ψ(t,u),x〉 is theLaplace transform of a probability distribution supported on K. It will turnout that this probability distribution corresponds to the non-central Wishartdistribution in the case of positive semidefinite matrices. The existence ofsuch affine diffusion processes then follows from the semiflow property of φand ψ, which yields the Kolmogorov-Chapman equation for the transitionprobabilities and thus the Markov property.

We start by establishing explicit solutions for the Riccati equations asso-ciated to the parameter set (α, δα, 0, 0, 0, 0, 0) with δ ∈ R+.

Lemma 3.4.1. Let α ∈ K and δ ∈ R+. Consider the following system ofRiccati differential equations for u ∈ K and t ∈ R+

∂ψ(t, u)

∂t= −2P (ψ(t, u))α, ψ(0, u) = u ∈ K, (3.38)

∂φ(t, u)

∂t= 〈δα, ψ(t, u)〉, φ(0, u) = 0. (3.39)

Then the solution is given by

ψ(t, u) = (u−1 + 2tα)−1, (3.40)

φ(t, u) =δ

2ln det

(e+ 2tP (

√α)u). (3.41)

Moreover, u 7→ ψ(t, u) and u 7→ φ(t, u) are continuous at u = 0 for all t ∈ R+

with φ(t, 0) = 0 and ψ(t, 0) = 0.

3.4. Construction of Affine Processes on Symmetric Cones 87

Proof. Usingd

dt(x+ tv)−1 = −P ((x+ tv)−1)v,

which follows from Proposition A.2.2 (iii), one easily verifies that ψ(t, u) givenby (3.40) satisfies (3.38). Concerning φ(t, u), let us first show that

det(u) det(u−1 + 2tα) = det(e+ 2tP (

√α)u). (3.42)

Indeed, we have by Proposition A.2.2 (i), (iv) and Proposition A.1.3

det(u) det(u−1 + 2tα) = det(u) det(P (√u−1)

(e+ 2tP (

√u)α))

= det(u) det(u−1) det(e+ 2tP (

√u)α)

= det(uu−1) det(e+ 2tP (

√u)α)

= det(e+ 2tP (

√u)α)

= det(e+ 2tP (

√α)u).

The last equality follows again from Proposition A.2.2 (iv), which implies

det(P (√u)α) = det(u) det(α) = det(P (

√α)u).

Hence φ(t, u) can be written as

φ(t, u) =δ

2ln det(u) +

δ

2ln det(u−1 + 2tα). (3.43)

Using expression (3.43) for φ(t, u) and ∇ ln detx = x−1 yields

∂φ(t, u)

∂t=

⟨δ

2(u−1 + 2tα)−1, 2α

⟩= 〈δα, ψ(t, u)〉,

and shows that (3.41) solves (3.39).

Let now ψ(t, u) and φ(t, u) be given by (3.38) and (3.39) and consider

Lδ,α,xt (u) := e−φ(t,u)−〈ψ(t,u),x〉 = det(e+ 2tP (

√α)u)− δ

2 e−〈(u−1+2tα)−1,x〉

for u ∈ K. We shall now prove that for every t ∈ R+ and α, x ∈ K

u 7→ Lδ,α,xt (u) (3.44)

is the Laplace transform of a probability measure on K if δ ≥ d(r − 1).In the case K = S+

r , this is implied by Letac and Massam [2004, Proposition


3.2], which asserts that Lδ,α,xt corresponds to the Laplace transform of the non-central Wishart distribution. The proof is based on the density function of thisdistribution, which exists for δ > (r− 1) and α ∈ S++

r . As such a result is notavailable for general symmetric cone, we choose a different approach. However,in Section 3.5 we establish the form of the density corresponding to Lδ,α,xt ,which then yields a generalization of the non-central Wishart distribution onsymmetric cones.

In order to prove (3.44), the following lemma states that it is enough toconsider α = e and t = 1

2.

Lemma 3.4.2. Let α ∈ K and δ ≥ d(r − 1) and t ∈ R+ be fixed. Supposethat for every x ∈ K

Lδ,e,x12

(u) = det (e+ u)−δ2 e−〈(u

−1+e)−1,x〉

is the Laplace transform of a probability measure on K. Then this assertionalso holds true for Lδ,α,xt .

Proof. Let t > 0 and x be fixed. Suppose first that α ∈ K. We denote by Xthe random variable whose Laplace transform is given by Lδ,e,x1

2

. Consider now

another random variable Y defined by Y = 2tP (√α)X and let y = 2tP (

√α)x.

We then have by Proposition A.2.2 (i),

E[e−〈u,Y 〉

]= E

[e−〈2tP (

√α)u,X〉

]= det

(e+ 2tP (

√α)u)− δ

2 e−〈((2tP (√α)u)−1+e)−1,x〉

= det(e+ 2tP (

√α)u)− δ

2 e−〈2tP (√α)(u−1+2tP (

√α)e)−1, 1

2tP(√α−1)y〉

= det(e+ 2tP (

√α)u)− δ

2 e−〈(u−1+2tα)−1,y〉

= Lδ,α,yt .

Since x was arbitrary and as P (√α) is an automorphism on K, Lδ,α,yt is a

Laplace transform of a probability distribution for every y ∈ K. For degen-erate α the assertion is a consequence of Levy’s continuity theorem, since

we have locally uniform convergence on K of Lδ,α+ en,x

t to Lδ,α,xt , which is acontinuous function at 0.

Due to this lemma, it is sufficient to consider det (e+ u)−δ2 e−〈(u

−1+e)−1,x〉.In the following we fix a Jordan frame p1, . . . , pr and denote for all m ∈

1, . . . , r

em :=m∑i=1

pi, V (m) := V (em, 1),


where V (em, 1) = x ∈ V : em x = x (see Section A.1.2). Moreover,K(m) corresponds to the symmetric cone associated with the subalgebra V (m)

of rank m, that is, the relative interior of the set of squares in V (m). For ageneral indempotent c ∈ V (m), K

(m)c denotes the symmetric cone associated

with the subalgebra V (m)(c, 0).

The following lemmas are based on an idea of Ahdida and Alfonsi [2010].

Lemma 3.4.3. Let m ∈ 1, . . . , r. Fix δ ≥ d(m − 1), t ∈ R+ and x ∈K(m). Suppose that Lδ,em−1,x

t and Lδ,pm,xt are Laplace transforms of probabilitymeasures on K(m). Then the same holds true for Lδ,em,xt .

Proof. Define φi(t, u) and ψi(t, u) for i = 1, 2 and u ∈ K by

φ1(t, u) =δ

2ln det(e+ 2tP (pm)u), ψ1(t, u) = (u−1 + 2tpm)−1,

φ2(t, u) =δ

2ln det (e+ 2tP (em−1)u) , ψ2(t, u) =

(u−1 + 2tem−1

)−1.

Hence Lδ,pm,xt = e−φ1(t,u)−〈ψ1(t,u),x〉 and Lδ,em−1,xt = e−φ2(t,u)−〈ψ2(t,u),x〉. We now

claim that

Lδ,em,xt (u) = e−φ(t,u)+〈ψ(t,u),x〉 = det (e+ 2tP (em)u)−δ2 e−〈(u

−1+2tem)−1,x〉

is given by e−φ1(t,u)−φ2(t,ψ1(t,u))−〈ψ2(t,ψ1(t,u)),x〉. Indeed, we have

ψ2(t, ψ1(t, u)) =

(((u−1 + 2tpm

)−1)−1

+ 2tem−1

)−1

=(u−1 + 2tpm + 2tem−1

)−1

= (u−1 + 2tem)−1 = ψ(t, u),

and by using (3.42),

e−φ1(t,u)−φ2(t,ψ1(t,u))

= det(e+ 2tP (pm)u)−δ2 det

(e+ 2tP (em−1) (u−1 + 2tpm)−1

)− δ2

= det(u)−δ2 det(u−1 + 2tpm)−

δ2 det((u−1 + 2tpm)−1)−

δ2

× det(((u−1 + 2tpm)−1)−1 + 2tem−1

)− δ2

= det(u)−δ2 det(u−1 + 2tem)−

δ2

= det (e+ 2tP (em)u)−δ2 = e−φ(t,u).


Denoting the probability measures associated with Lδ,pm,xt and Lδ,em−1,xt by

p1t (x, dξ) and p2

t (x, dξ), we thus have

Lδ,em,xt (u) = e−φ1(t,u)−φ2(t,ψ1(t,u))−〈ψ2(t,ψ1(t,u)),x〉

=

∫K(m)

∫K(m)

e−〈u,ξ〉p1t (ξ, dξ)p

2t (x, dξ),

implying that Lδ,em,xt is the Laplace transform of the probability measure∫K(m)

p1t (ξ, ·)p2

t (x, dξ).

The following lemma states a decomposition of x into the sum of twoelements of smaller rank.

Lemma 3.4.4. Let x ∈ K(m) for some m ∈ 2, . . . , r and c an idempotent inV (m) of rank k, where k ∈ 1, . . . ,m− 1. Consider the Peirce decompositionof x in V (m) with respect to c, that is, x = x1 +x 1

2+x0 and denote the inverse

of x1 in V (m)(c, 1) by x−11 .3 Then x can be decomposed into x = y + z, where

y = x1 + x 12

+ P (x 12)x−1

1 ∈ K(m), (3.45)

z = x0 − P (x 12)x−1

1 ∈ K(m)c ⊂ V (m)(c, 0). (3.46)

Proof. By the Peirce multiplication rule (A.6), we have

P

(V (m)

(c,

1

2

))V (m)(c, 1) ⊂ V (m)(c, 0),

whence P (x 12)x−1

1 ∈ K(m)c , from which we can deduce that y ∈ K(m). Us-

ing Massam and Neher [1997, Proposition 3.3.1 c)], we can write

((x−1)0)−1 = x0 − P (x 12)x−1

1 ,

where ((x−1)0)−1 is the inverse of (x−1)0 in V (m)(c, 0). As this lies in K(m)c ,

the assertion is proved.

Remark 3.4.5. (i) In the case V (m) = Sm, the matrix(Ik 00 0

)3Note that this is well-defined, since we assume x ∈ K(m).


is an idempotent of rank k. The above decomposition then correspondsto (

x1 x>12

x12 x0

)=

(x1 x>12

x12 x12x−11 x>12

)+

(0 00 x0 − x12x

−11 x>12

).

(ii) In the case of the two-dimensional Lorentz cone with m = r = 2 andk = 1, the Peirce decomposition with respect to an idempotent of rank1 is given by x = x1 + x0, since dim(V (c, 1

2)) = 0. Thus the above

decomposition reduces to y = x1 and z = x0.

In the following we show that, for integer values k and particular elementsy of form (3.45), Ldk,em−c,y1

2

can be recognized as the Laplace transform of

a quadratic function of a Gaussian random variable supported on V (c, 12),

where c denotes some idempotent specified below. Before stating the lemmafor general irreducible symmetric cones, let us examplify the assertion forV (m) = Sm. Let y ∈ S+

m such that its Peirce decomposition with respect to(Ik 00 0

)is given by (

y1 y>12

y12 y12y−11 y>12

).

Consider for some z ∈ V (m)(Ik,12) the following linear transformation of sym-

metric matrices

τ(z) : Sm → Sm,

(x1 x>12

x12 x0

)7→(

x1 x1z> + x>12

zx1 + x12 zx1z> + x12z

> + zx>12 + x0

).

Let now Z be a standard Gaussian random variable on V (m)(Ik,12). We then

prove that the Laplace transform of the S+m-valued random variable

τ

(2

√y−1

1 Z)y =

(y1

√y1Z

> + y>12

Z√y1 + y12 q(Z)

)with

q(Z) =

(Z + y12

√y−1

1

)(Z + y12

√y−1

1

)>is given by Lk,Im−Ik,y1

2

, where we identify Z with an element in Rm−k×k.

The generalization of the map τ(z) to Euclidean Jordan algebras is theso-called Frobenius transform introduced in Definition A.2.3. This transformis used to prove the following lemma.


Lemma 3.4.6. Let c be an idempotent in V (m) of rank k and tr(c) = k,where m ∈ 2, . . . , r and k ∈ 1, . . . ,m − 1. Moreover, let y ∈ K(m) be ofform (3.45), that is,

y = y1 + y 12

+ y0 = y1 + y 12

+ P (y 12)y−1

1

for some y1 ∈ V (m)(c, 1) and y 12∈ V (m)(c, 1

2). Then

u 7→ Ldk,em−c,y12

(u) = det(e+ P (em − c)u)−dk2 e−〈(u

−1+(em−c))−1,y〉, u ∈ K,(3.47)

is the Laplace transform of a probability distribution on K(m).

Proof. We show that there exists a random variable X taking values in K(m)

such that

E[e−〈u,X〉

]= det(e+ P (em − c)u)−

dk2 e−〈(u

−1+(em−c))−1,y〉.

By Faraut and Koranyi [1994, Proposition VI.4.1], the function

σ : V (m)(em − c, 1)→ L(V (m)

(c,

1

2

)),

defined by

σ(x)ξ = 2x ξ = 2L(x)ξ

is a self-adjoint representation of V (m)(em − c, 1) on V (m)(c, 12) with

σ(em − c) = I,

where I denotes the identity. In particular, since V (m)(em− c, 1) = V (m)(c, 0),

there is a K(m)c -valued quadratic form Q : V (m)(c, 1

2) → K

(m)c ⊂ V (m)(c, 0)

given by

〈σ(x)ξ, ξ〉 = 〈2x ξ, ξ〉 = 〈x, 2ξ2〉 = 〈x,Q(ξ)〉, x ∈ V (m)(c, 0),

which implies that Q(ξ) = 2L(em − c)ξ2. Let now Z be a Gaussian randomvariable Z ∼ N(0, I) on V (m)(c, 1

2) and define theK(m)-valued random variable

X by

X = τ

(2

√y−1

1 Z)y,


where τ denotes the Frobenius transform, as introduced in Definition A.2.3.According to Faraut and Koranyi [1994, Lemma VI.3.1], the Peirce decompo-sition of X with respect to c is then given by

X1 = y1,

X 12

= 4L

(√y−1

1 Z)y1 + y 1

2,

X0 = 2L(em − c)L(

2

√y−1

1 Z)2

y1 + 4L(em − c)L(√

y−11 Z

)y 1

2+ y0.

Denoting by L(em)u = u1 + u 12

+ u0 the Peirce decomposition of L(em)u

with respect to c in V (m), we obtain due to the orthogonality of V (c, λ), forλ = 1, 1

2, 0 and the fact that X takes values in V (m)

〈u,X〉 = 〈u1, y1〉+

⟨u 1

2, 4L

(√y−1

1 Z)y1 + y 1

2

⟩+

⟨u0, 2L

(2

√y−1

1 Z)2

y1 + 4L

(√y−1

1 Z)y 1

2+ y0

⟩,

= 〈u1, y1〉+⟨u 1

2, y 1

2

⟩+ 〈u0, y0〉

+ 4

⟨L

(√y−1

1

)L(y1)u 1

2, Z

⟩+ 4

⟨L

(√y−1

1

)L(y 1

2)u0, Z

⟩+

1

2〈u0, Q(Z)〉 .

(3.48)

Here, the last equality follows from

2L(em − c)L(

2

√y−1

1 Z)2

y1 = L(em − c)Z2 =1

2Q(Z),

which is a consequence of Massam and Neher [1997, Equation 3.5, 3.6].For notational reasons we define

η := L

(√y−1

1

)(L(y1)u 1

2+ L(y 1

2)u0

)=

1

2L(u 1

2

)√y1 +L(u0)L

(√y−1

1

)y 1

2,

where we use Faraut and Koranyi [1994, Proposition II.1.1] and again Massamand Neher [1997, Equation 3.5]. Let us now compute

E[e−

12〈u0,Q(Z)〉−4〈η,Z〉

].


Denoting the dimension of V (m)(c, 12) by N := dk(m− k) and using

σ(em − c) = I, we have

E[e−

12〈u0,Q(Z)〉−4〈η,Z〉

]=

∫V (m)(c, 1

2)

1

(2π)N2

e−12〈u0,Q(z)〉−4〈η,z〉e−

12〈σ(em−c)z,z〉dz

= Det(σ(u0 + (em − c)))−12

×∫V (m)(c, 1

2)

1

(2π)N2

e−12〈(σ(u0+em−c)z,z〉−4〈η,z〉Det(σ(u0 + (em − c)))

12dz

= det(u0 + (em − c))−N

2(m−k) e12〈σ((u0+em−c)−1)4η,4η〉

= det(u0 + (em − c))−dk2 e16〈L((u0+(em−c))−1)η,η〉.

Here, we use the moment generating function for the normal distribution with

covariance σ((u0+(em−c))−1), the relation Det(σ(x)) = det(x)N

m−k (see Farautand Koranyi [1994, Proposition IV.4.2]) and the fact that σ(x) = 2L(x) in ourcase. The linear map L is restricted to V (m)(c, 0) = V (m)(em − c, 1) such thatthe inverse is well-defined. Note that det(u0 + (em − c)) = det(P (em − c)u +(em− c)) = det(P (em− c)u+ e). Due to (3.48), it thus only remains to prove

− 〈u1, y1〉 − 〈u 12, y 1

2〉 − 〈u0, y0〉+ 16〈L((u0 + (em − c))−1)η, η〉

= −〈(u−1 + (em − c))−1, y〉. (3.49)

By the definition of η and the fact that

y0 = P (y 12)y−1

1 = L(em − c)L(

2

√y−1

1 y 12

)2

e,

we obtain using several times Massam and Neher [1997, Equation 3.5]

− 〈u1, y1〉 − 〈u 12, y 1

2〉 − 〈u0, y0〉+ 16〈L((u0 + (em − c))−1)η, η〉

= 〈2L(u 12)L(u 1

2)(u0 + (em − c))−1 − u1, y1〉

+ 〈2L(u 12)L(u0)(u0 + (em − c))−1 − u 1

2, y 1

2〉

+ 〈L(u0)L(u0)(u0 + (em − c))−1 − u0, y0〉.

On the other hand we can express −L(em)(u−1 + (em − c))−1 as

− L(em)(u−1 + (em − c))−1 = P (u0 + u 12)(u0 + (em − c))−1 − L(em)u

= (P (u 12) + 2P (u0, u 1

2) + P (u0))(u0 + (em − c))−1 − L(em)u.


The above claim (3.49) is then a consequence of the following identities (Mas-sam and Neher [1997, Equation 3.7] for the first one)

P (u 12)(u0 + (em − c))−1 = 2L(c)L(u 1

2)L(u 1

2)(u0 + (em − c))−1,

P (u0, u 12)(u0 + (em − c))−1 = L(u 1

2)L(u0)(u0 + (em − c))−1,

P (u0)(u0 + (em − c))−1 = L(u0)L(u0)(u0 + (em − c))−1,

whence the assertion is proved.

Remark 3.4.7. Note that in the case of the two-dimensional Lorentz coney = y1 and (3.47) is reduced to e−〈u1,y1〉, which is simply the Laplace transformof the Dirac measure at y1.

Using the above lemmas, we are now prepared to prove the crucial proposi-tion which shall allow us to establish the existence of affine diffusion processes.

Proposition 3.4.8. Let m ∈ 1, . . . , r. Then, for all δ0 ≥ 0 and x ∈ K(m),

u 7→ Lδ0+d(m−1),em,x12

(u), u ∈ K,

is the Laplace transform of a probability measure on K(m).

Proof. We proceed by induction on m. Let us start with m = 1. Then, for allδ0 ≥ 0 and x ∈ K(1) = R+p1, Lδ0,p1,x

12

is a Laplace transform of a probability

distribution on K(1) = R+p1. Indeed,

Lδ0,p1,x12

(u) = det (e+ P (p1)u)−δ02 e−⟨(u−1+p1)

−1,x⟩

=1

(1 + s)δ02

exp

(− sa

1 + s

),

where s ≥ 0 is defined through P (p1)u = sp1 and a ≥ 0 by x = ap1. Here, weuse the fact that P (p1) is the orthogonal projection on V (1).Noticing that

s 7→ 1

(1 + s)δ02

exp

(− sa

1 + s

)(3.50)

is the Laplace transform of the image of the non-central chi-square distribution(with δ0 degrees of freedom and non-centrality parameter depending on a)under some positive linear map (see, e.g., Letac and Massam [2004, Definition1.1]), implies the above claim.


As induction hypothesis we now assume that for all δ0 ≥ 0 and z ∈ K(m−1),Lδ0+d(m−2),em−1,z

12

is a Laplace transform of a probability distribution onK(m−1).

For the induction step from m− 1 to m, we prove that for all x ∈ K(m),

Lδ0+d(m−1),em−1,x12

and Lδ0+d(m−1),pm,x12

are Laplace transforms of probability distributions on K(m). An applicationof Lemma 3.4.3 then yields the assertion.

Let now x ∈ K(m) be fixed. By Lemma 3.4.4, we can decompose x withrespect to pm into x = y+z, where y is of form (3.45) and z ∈ K(m)

pm = K(m−1).

We start by considering Lδ0+d(m−1),em−1,x12

, which can therefore be written as

Lδ0+d(m−1),em−1,x12

= det (e+ P (em−1)u)−δ0+d(m−1)

2 e−⟨(u−1+em−1)

−1,x⟩

= det (e+ P (em−1)u)−δ0+d(m−2)

2 e−⟨(u−1+em−1)

−1,z⟩

(3.51)

× det (e+ P (em−1)u)−d2 e−⟨(u−1+em−1)

−1,y⟩. (3.52)

By our induction hypothesis, the first term (3.51) is the Laplace transform ofa probability distribution on K(m−1). Due to Lemma 3.4.6, the same holdstrue for the second term (3.52), but with support on K(m). Hence the productof these two terms is again the Laplace transform of a probability distributionon K(m). Let us now turn to Lδ0+d(m−1),pm,x

12

, which by Lemma 3.4.4 can be

written as

Lδ0+d(m−1),pm,x12

= det(e+ P (pm)u)−δ0+d(m−1)

2 e−〈(u−1+pm)−1,x〉

= det(e+ P (pm)u)−d(m−1)

2 e−〈(u−1+pm)−1,y〉 (3.53)

× det(e+ P (pm)u)−δ02 e−〈(u

−1+pm)−1,z〉, (3.54)

where z ∈ K(m)em−1 , which is the cone associated with V (em−1, 0), and y is of

form (3.45). Note that the corresponding idempotent is now em−1. Due toLemma 3.4.6, the first term (3.53) corresponds again to the Laplace transformof a probability distribution on K(m), and the second one (3.54) to the non-

central chi-square distribution supported on the one-dimensional cone K(m)em−1 ,

similar as in (3.50), where p1 is simply replaced by pm. Hence, by Lemma 3.4.3,

the assertion is proved for all x ∈ K(m). For degenerate x, Lδ0+d(m−1),em,x+ en

12

converges locally uniformly on K to Lδ0+d(m−1),em,x12

. Due to the continuity

of u 7→ Lδ0+d(m−1),em,x12

(u) at 0 and Levy’s continuity theorem, the claim also

holds for x ∈ ∂K(m).


The following corollary is an immediate consequence of the above propo-sition.

Corollary 3.4.9. Let ψ(t, u) and φ(t, u) be given by (3.38) and (3.39) andconsider

Lδ,α,xt (u) = e−φ(t,u)+〈ψ(t,u),x〉 = det(e+ 2tP (

√α)u)− δ

2 e−〈(u−1+2tα)−1,x〉

for u ∈ K. If δ ≥ d(r−1), then, for every t ∈ R+ and α, x ∈ K, u 7→ Lδ,α,xt (u)is the Laplace transform of a probability measure on K.

Proof. From Proposition 3.4.8 it immediately follows that, for every δ ≥d(r − 1) and x ∈ K, Lδ,e,x1

2

is the Laplace transform of a probability measure

on K. An application of Lemma 3.4.2 then yields the assertion for generalparameters.

Using the knowledge that Lδ,α,xt is the Laplace transform of a probabilitydistribution on K, we can finally prove existence of affine diffusion processesassociated with the particular parameter set (α, δα, 0, 0, 0, 0, 0), where α ∈ Kand δ ≥ d(r − 1).

Proposition 3.4.10. Let (α, δα, 0, 0, 0, 0, 0) be an admissible parameter set,that is, α ∈ K and δ ≥ d(r− 1). Then there exists a unique affine process onK such that (2.1) holds for all (t, u) ∈ R+ ×K, where φ(t, u) and ψ(t, u) aregiven in Lemma 3.4.1.

Proof. By Proposition 3.4.9 we have for every (t, x) ∈ R+ ×K the existenceof a probability measure on K with Laplace-transform e−φ(t,u)−〈ψ(t,u),x〉, whereφ(t, u) and ψ(t, u) are specified in Lemma 3.4.1. The Chapman-Kolmogorovequations hold in view of the flow property of φ and ψ, whence the assertionfollows.

Remark 3.4.11. In the case of positive semidefinite matrices, Bru [1991] hasshown existence and uniqueness for the process

dXt = δIr +√XtdWt + dW>

√Xt, X0 = x,

if δ > r − 1 and x with distinct eigenvalues (see Bru [1991, Theorem 2 andSection 3]). This process corresponds to the parameter set (Ir, δIr, 0, 0, 0, 0, 0)on the cone S+

r (see Theorem 3.7.2 below). Note that, since the Peirce in-variant d equals 1 in this case, δ > r − 1 is a stronger assumption than whatwe require on δ. Actually, Bru [1991] establishes existence and uniqueness ofsolutions also for δ = 1, . . . , r− 1. But these are degenerate solutions, as theyare only defined on lower dimensional subsets of the boundary of S+

r (see Bru[1991, Corollary 1]).


3.4.2 Existence of Affine processes on Symmetric cones

In Proposition 3.4.10 and Proposition 2.5.3 we have proved existence of affinediffusion processes with a particular constant drift parameter and existence ofpure affine jump processes. In order to establish existence of affine processeson symmetric cones for any admissible parameter set, we now combine therespective Riccati equations to show that

e−φ(t,u)−〈ψ(t,u),x〉 = limN→∞

[P 2

tNP 1

tN

]Ne−〈u,x〉,

is the Laplace transform of a probability distribution on K for any admissibleparameter set. Here, P i, i = 1, 2, denote the respective semigroups of thediffusion process and the pure jump process.

Given an admissible parameter set (α, b, B, c, γ,m, µ), let us therefore con-sider the following two systems of Riccati ODEs:

∂tψ1(t, u) = R1(ψ1(t, u)) = −2P (ψ1(t, u))α,

∂tφ1(t, u) = F1(ψ1(t, u)) = 〈δα, ψ1(t, u)〉,∂tψ2(t, u) = R2(ψ2(t, u)) = B>(ψ2(t, u)) + γ

−∫K

(e−〈ξ,ψ2(t,u)〉 − 1 + 〈χ(ξ), ψ2(t, u)〉

)µ(dξ),

∂tφ2(t, u) = F2(ψ2(t, u)) = 〈b− δα, ψ2(t, u)〉+ c−∫K

(e−〈ξ,ψ2(t,u)〉 − 1

)m(dξ),

(3.55)

where we set δ = d(r − 1). The original Riccati equations corresponding tothe parameter set (α, b, B, c, γ,m, µ) are then given by

∂tψ(t, u) = R(ψ(t, u)) = R1(ψ(t, u)) +R2(ψ(t, u)), ψ(0, u) = u, (3.56)

∂tφ(t, u) = F (ψ(t, u)) = F1(ψ(t, u)) + F2(ψ(t, u)), φ(0, u) = 0. (3.57)

Let us remark that, due to Theorem 2.4.3, there exists a global uniquesolution to (3.56)-(3.57) for every u ∈ K, which remains in K for all t ∈ R+.

Lemma 3.4.12. Let φi, ψi, i = 1, 2, be defined by (3.55) and let u ∈ K andt ≥ 0 be fixed. Define recursively for each N ∈ N and n ∈ 0, . . . , N

y0(u) := u, w0(u) := 0,

yn(u) := ψ2(τ, ψ1(τ, yn−1)), wn(u) := φ1(τ, yn−1) + φ2(τ, ψ1(τ, yn−1) + wn−1,

where τ = tN

. Then

ψ(t, u) = limN→∞

yN(u) and φ(t, u) = limN→∞

wN(u),

where φ and ψ are given by (3.56)-(3.57).


Proof. Let us first remark that the limits are well-defined, since we have exis-tence of global solutions of (3.56)-(3.57) by Theorem 2.4.3. In order to proveconvergence of this splitting scheme, let us first calculate the local errors of theapproximations for φ and ψ for a given step size τ = t

Nwith N ∈ N fixed. An

estimate of the global error is then obtained by transporting the local errorsto the final point t and adding them up, as it is done in Hairer, Nørsett, andWanner [1993, Theorem 3.6]. Following Hairer et al. [1993, Chapter II.3], letus define the increment functions Φψ and Φφ by

yn(u) = yn−1(u) + τΦψ(yn−1(u), τ),

wn(u) = wn−1(u) + τΦψ(yn−1(u), τ).

Using a Taylor expansions at τ = 0, we obtain due to the analyticity of R1, R2

and F1, F2 on K for y ∈ K

Φψ(y, τ) = R2(y) +R1(y)

+1

2τ(DR2(y)R2(y) + 2DR2(y)R1(y) +DR1(y)R1(y))

+O(τ 2),

Φφ(y, τ) = F1(y) + F2(y)

+1

2τ(〈DF2(y), R2(y)〉+ 2〈DF2(y), R1(y)〉+ 〈DF1(y), R1(y)〉)

+O(τ 2).

Hence the local errors satisfy by another Taylor expansion of ψ and φ

‖ψ(t+ τ, u)− ψ(t, u)− τΦψ(ψ(t, u), τ)‖

=1

2τ 2‖DR1(ψ(t, u))R2(ψ(t, u))−DR2(ψ(t, u))R1(ψ(t, u))‖+O(τ 3)

≤ Cψτ2,

|φ(t+ τ, u)− φ(t, u)− τΦφ(ψ(t, u), τ)|

=1

2τ 2|〈DF1(ψ(t, u)), R2(ψ(t, u))〉 − 〈DF2(ψ(t, u)), R1(ψ(t, u))〉|+O(τ 3)

≤ Cφτ2.

Since R1, R2 and F1, F2 are analytic on K, the following Lipschitz conditionsfor some constants Λψ,Λφ are satisfied in a neighborhood of the solution

‖Φψ(z, τ)− Φψ(y, τ)‖ ≤ Λψ‖z − y‖, |Φφ(z, τ)− Φφ(y, τ)| ≤ Λφ‖z − y‖.


Moreover, by Theorem 2.4.3, ψ(t, u) ∈ K for all (t, u) ∈ R+× K such that wehave by Hairer et al. [1993, Theorem II.3.6]

‖ψ(t, u)− yN(u)‖ ≤ τCψΛψ

eΛψt−1,

‖φ(t, u)− wN(u)‖ ≤ τCφΛφ

eΛφt−1,

which converges by the definition of τ to 0 as N →∞.

We are now prepared to prove the main result of this section, which es-tablishes existence of affine processes on irreducible symmetric cones for anygiven admissible parameter set.

Theorem 3.4.13. Let (α, b, B, c, γ,m, µ) be an admissible parameter set. Thenthere exists a unique affine process on K, such that (2.1) holds for all (t, u) ∈R+ ×K, where φ(t, u) and ψ(t, u) are given by (2.9) and (2.10).

Proof. By Lemma 3.4.12, we have for each fixed t

e−φ(t,u)−〈ψ(t,u),x〉 = limN→∞

e−wN (u)−〈yN (u),x〉, u ∈ K.

For each N ∈ N, n ∈ 0, . . . , N and x ∈ K, u 7→ e−wn(u)−〈yn(u),x〉 is theLaplace transform of a probability distribution on K. Indeed, let us proceedby induction. For n = 0, e−〈u,x〉 is the Laplace transform of δx(dξ). Wenow suppose that for every x ∈ K, e−wn−1−〈yn−1,x〉 is the Laplace transformof a probability distribution µn−1(x, ·) on K. Due to Proposition 3.4.9 andProposition 2.5.2, u 7→ e−φi(τ,u)−〈ψi(τ,u),x〉, i = 1, 2, are Laplace transforms ofprobability measures supported on K, which we denote by piτ (x, dξ), i = 1, 2.Since we have

e−wn−〈yn,x〉 = e−wn−1e−φ1(τ,yn−1)−φ2(τ,ψ1(τ,yn−1))−〈ψ2(τ,ψ1(τ,yn−1)),x〉

=

∫K

∫K

e−wn−1−〈yn−1,ξ〉p1τ (ξ, dξ)p

2τ (x, dξ),

=

∫K

∫K

∫K

e−〈u,z〉µn−1(ξ, dz)p1τ (ξ, dξ)p

2τ (x, dξ),

e−wn−〈yn,x〉 is the Laplace transform of the probability distribution given by

µn(x, ·) =

∫K

∫K

µ(ξ, ·)p1τ (ξ, dξ)p

2τ (x, dξ).

3.5. Wishart Distribution 101

As u 7→ e−φ(t,u)−〈ψ(t,u),x〉 is continuous at 0 and the limit of a sequence ofLaplace transforms of probability distributions supported on K, Levy’s conti-nuity theorem implies that u 7→ e−φ(t,u)−〈ψ(t,u),x〉 is also the Laplace transformof a probability distribution on K.

Moreover, the Chapman-Kolmogorov equation holds in view of the semi-flow property of φ and ψ, which implies the assertion.

3.5 Wishart Distribution

In Proposition 3.4.9 we have seen that for every t ∈ R+ and α, x ∈ K

Lδ,α,xt (u) = det(e+ 2tP (

√α)u)− δ

2 e−〈(u−1+2tα)−1,x〉

is the Laplace transform of a probability measure on K if δ ≥ d(r− 1). In thecase of positive semidefinite matrices, this probability measure correspondsactually to the non-central Wishart distribution (see, e.g., Letac and Massam[2004]). For certain parameter values, namely δ > (r − 1) and α ∈ S++

r ,this distribution admits a density, which is explicitly known. We extend thisto general irreducible symmetric cones and establish the explicit form of theMarkov kernels corresponding to the affine diffusion processes, associated withthe parameter set (α, δα, 0, 0, 0, 0, 0), for δ > d(r − 1) and α ∈ K.

3.5.1 Central Wishart Distribution

We start by analyzing the case x = 0, which corresponds to the central Wishartdistribution (see, e.g., Letac and Massam [2004] or Massam and Neher [1997]).The following proposition extends the assertion of Corollary 3.4.9 to the setδ ∈ 0, d, . . . , d(r − 1) and states the form of the density in the case δ >d(r − 1) and α ∈ K.

Proposition 3.5.1. Let φ(t, u) be given by (3.39) and consider

Lδ,αt (u) := e−φ(t,u) = det(e+ 2tP (

√α)u)− δ

2 .

If δ belongs to the set

G = 0, d, . . . , d(r − 1) ∪ ]d(r − 1),∞[ ,

then, for every t ∈ R+ and α ∈ K, u 7→ Lδ,αt (u) is the Laplace transform of a

probability measure Wδ2,α

t on K. Moreover, if δ > d(r− 1) and if α ∈ K, then


Wδ2,α

t admits a density, which is given by

Wδ2,α

t (ξ) =1

ΓK(δ2

) det

(α−1

2t

) δ2

e−⟨α−1

2t,ξ⟩

det(ξ)δ2−nr , (3.58)

where ΓK denotes the Gamma function of K (see Faraut and Koranyi [1994,Section VII.1]).

Proof. By Faraut and Koranyi [1994, Theorem VII.3.1 and Proposition VII.2.3],

det(u)−δ2 is the Laplace transform of a positive measure if and only if δ ∈ G.

This is equivalent to the fact that det(u)−δ2 is a function of positive type (see,

e.g., Faraut and Koranyi [1994, page 136]), that is,

N∑i,j=1

det(ui + uj)− δ

2 cicj ≥ 0

for all choices of u1, . . . , uN ∈ K and complex numbers c1, . . . , cN . For every

t ∈ R+, Lδ,αt (u) = det (e+ 2tP (√α)u)

− δ2 is therefore also a function of positive

type and hence the Laplace transform of a positive measure if δ ∈ G. SinceLδ,αt (u + v) ≤ Lδ,αt (u) for all u, v ∈ K, the measure is supported on K. AsLδ,αt (u) is bounded and Lδ,αt (0) = 1, the measure is actually a probabilitymeasure.

Concerning the second assertion, we have by Faraut and Koranyi [1994,Corollary VII.1.3] and Proposition A.2.2 (iv)∫K

e−⟨u+α−1

2t,ξ⟩

det(ξ)δ2−nr dξ = ΓK

(δ

2

)det

(u+

α−1

2t

)− δ2

= ΓK

(δ

2

)det

(P (√α−1)

2t

(2tP (√α)u+ e

))− δ2= ΓK

(δ

2

)det

(α−1

2t

)− δ2

det(e+ 2tP (

√α)u)− δ

2 .

The definition of the density of Wδ2,α

t then yields the assertion.

Remark 3.5.2. (i) Analogous to the cone of positive semidefinite matrices,one can define the central Wishart distribution W p,σ with shape param-eter

p ∈ G =

0,d

2, . . . ,

d(r − 1)

2

∪]d(r − 1)

2,∞[


and scale parameter σ ∈ K on a symmetric cone by its Laplace transformwhich takes the form stated in Proposition 3.5.1, that is,∫

K

e−〈u,ξ〉W p,σ(dξ) = det(e+ P (

√σ)u)−p

.

(see, e.g., Massam and Neher [1997, Corollary 3]).

(ii) If p = k d2, k ∈ 1, . . . , r − 1, then the Wishart distribution is supported

on the set of elements in K which are precisly of rank k. This is aconsequence of Faraut and Koranyi [1994, Proposition VII.2.3]. In thecase of S+

r , this corresponds to the distribution of∑r

i=1 ZiZ>i , where Zi

are standard Gaussian random variables in Rr.

3.5.2 Non-central Wishart distribution

In order to formulate Proposition 3.5.4 below, where we establish the form ofthe density function of the non-central Wishart distribution on a symmetriccone, let us introduce the so-called zonal polynomials (see Faraut and Koranyi[1994, Section XI.3, p. 234]).

Definition 3.5.3 (Zonal Polynomials). For each multi-index

m = (m1, . . . ,mr) ∈ Nr

with length |m| := m1 + · · ·+mr, we consider the generalized power functiondefined by

∆m(ξ) = ∆1(ξ)m1−m2∆2(ξ)m2−m3 . . .∆r(ξ)mr ,

where ∆i denotes the principal minors corresponding to the Jordan subalgebrasV (j) = V (p1 + . . . + pj, 1) with p1, . . . , pr some fixed Jordan frame. The mth

zonal polynomial Zm is now defined by

Zm(ξ) = ωm

∫O∈O

∆m(Oξ)dO,

where dO is the normalized Haar measure on O and O = G ∩ O(V ), whereG is the connected component of the identity in the automorphism group ofK and O(V ) the orthogonal group of V . Moreover, ωm denotes some positivenormalizing constant.

We now state for α ∈ K and δ > d(r − 1) the form of the density whoseLaplace transform is given by Lδ,α,xt . To this end we generalize a result by Letacand Massam [2004] on the density function of the non-central Wishart distri-bution on positive semidefinite matrices to symmetric cones.


Proposition 3.5.4. Let ψ(t, u) and φ(t, u) be given by (3.38) and (3.39) andconsider

Lδ,α,xt (u) := e−φ(t,u)+〈ψ(t,u),x〉 = det(e+ 2tP (

√α)u)− δ

2 e−〈(u−1+2tα)−1,x〉

for u ∈ K. If δ > d(r − 1) and if α ∈ K, then Lδ,α,xt (u) is the Laplacetransform of a density, which is given by

Wδ2,α,x

t (ξ) = det

(α−1

2t

) δ2

e−⟨α−1

2t,ξ+x

⟩det(ξ)

δ2−nr (3.59)

×

(∑m≥0

Zm

(1

4t2P (√x)P (α−1) ξ

)|m|!ΓK

(m + δ

2

) ), (3.60)

where ΓK denotes the Gamma function of K (see Faraut and Koranyi [1994,Section VII.1]) and Zm the zonal polynomials introduced in Definition 3.5.3.

Proof. We apply similar arguments as in Letac and Massam [2004] to prove the

assertion. We first establish that Wδ2,α,x

t , as given in (3.59), is a well-definedpositive measure. Concerning the convergence of

∑m≥0

Zm

(1

4t2P (√x)P (α−1) ξ

)|m|!ΓK

(m + δ

2

) , (3.61)

we can estimate ΓK(m+ δ2) due to Faraut and Koranyi [1994, Theorem VII.1.1]

by

ΓK

(m +

δ

2

)≥ (2π)

n−r2 (min

z≥0Γ(z))r =: M,

where Γ denotes the Gamma function on R. This implies convergence of (3.61),since we have by Faraut and Koranyi [1994, Proposition XII.1.3(i)]

etr(ξ) =∑m≥0

Zm(ξ)

|m|!(3.62)

for every ξ ∈ K. Due to the definition of the zonal polynomials, in particularsince ∆m(ξ) > 0 for all ξ ∈ K \ 0, W δ

2,α,x is therefore a well-defined positive

measure. Let us now prove that u 7→ Lδ,α,xt (u) is the Laplace transform of

Wδ2,α,x

t . For each m ∈ Nr and each automorphism g, we have by Faraut and


Koranyi [1994, Lemma XI.2.3] and Proposition A.2.2 (iv)

∫K

e−⟨u+α−1

2t,ξ⟩

det(ξ)δ2−nrZm(gξ)dξ

= ΓK

(m +

δ

2

)det

(u+

α−1

2t

)− δ2

Zm

(g

(u+

α−1

2t

)−1)

= ΓK

(m +

δ

2

)det

(α−1

2t

)− δ2

det(e+ 2tP

(√α)u)− δ

2

× Zm

(g

(u+

α−1

2t

)−1).

(3.63)

If x is non-degenerate, P (√x)P (α−1) is an automorphism and plays the role

of g in our case. However, the above formula also holds true if x is degenerate.Indeed, let us approximate x by xn = x+ 1

ne. Since (tr(ξ))k =

∑|m|=k Zm(ξ)

for every k (see p. 235 in Faraut and Koranyi [1994]), we have for |m| = k

Zm

(P (√xn)P

(α−1)ξ)≤(tr(P(α−1)ξxn))k ≤ (tr (P (α−1

)ξx1

))k.

Dominated convergence then yields (3.63) also for degenerate x. By (3.62) weobtain

det(e+ 2tP

(√α)u) δ

2

∫K

e−〈u,ξ〉Wδ2,α,x

t dξ

= e−⟨α−1

2t,x⟩∑

m≥0

Zm

(1

4t2P (√x)P (α−1)

(u+ α−1

2t

)−1)

|m|!

= e

−⟨α−1

2t,x⟩

+

⟨1

4t2P(√x)P(α−1)

(u+α−1

2t

)−1,e

⟩.

(3.64)

Using P (z) = P (√z)P (

√z), Proposition A.2.2 (ii) and Faraut and Koranyi

[1994, Exercise II.5 (c)], which asserts (z+e)−1−e = −(z−1+e)−1 for invertible


elements z, z + e and z−1 + e, we get

−⟨α−1

2t, x

⟩+

⟨1

4t2P(√

x)P(α−1)(

u+α−1

2t

)−1

, e

⟩

=

⟨−e+

1

2tP(√

α−1)(

u+α−1

2t

)−1

,1

2tP(√

α−1)x

⟩

=

⟨−e+

(2tP

(√α)u+ e

)−1,

1

2tP(√

α−1)x

⟩= −

⟨((2tP

(√α)u)−1

+ e)−1

,1

2tP(√

α−1)x

⟩= −

⟨2tP

(√α) (u−1 + 2tP

(√α)e)−1

,1

2tP(√

α−1)x

⟩= −

⟨(u−1 + 2tα)−1, x

⟩= −〈ψ(t, u), x〉 .

This proves that u 7→ Lδ,α,xt (u) is the Laplace transform of the density givenin (3.59).

Remark 3.5.5. The explicit form of the Markov kernels corresponding to theaffine diffusion processes associated with the parameter set (α, δα, 0, 0, 0, 0, 0),

for δ > d(r − 1) and α ∈ K, is thus given by Wδ2,α,x

t .

3.6 Relation to Infinitely Divisible Distribu-

tions

Let P be the set of all families of probability measures (Px)x∈K on the canonicalprobability space (Ω,F) such that (X, (Px)x∈K) is a stochastically continuousMarkov processes on K with Px[X0 = x] = 1 for all x ∈ K. For two probabilitymeasures P,Q on (Ω,F) the convolution P∗Q is defined as the push-forward ofthe product measure P×Q under the map (ω, ω′) 7→ ω+ω′ : (Ω×Ω,F⊗F)→(Ω,F).

Definition 3.6.1. An element (Px)x∈K ∈ P is called

(i) infinitely decomposable if, for each k ≥ 1, there exists (P(k)x )x∈K ∈ P

such thatPx(1)+···+x(k) = P(k)

x(1) ∗ · · · ∗ P(k)

x(k) .

3.6. Relation to Infinitely Divisible Distributions 107

(ii) infinitely divisible if the one-dimensional marginal distributions PxX−1t

are infinitely divisible for all (t, x) ∈ R+ ×K.

In Duffie et al. [2003, Theorem 2.15] it was shown that affine processeson Rm

+ × Rn are infinitely decomposable Markov processes, and vice versa.In fact, this property was the core for the existence proof of affine processesin Duffie et al. [2003]. As we have seen in the previous section, the situation onsymmetric cones is different, since the non-central Wishart distributions are nolonger infinitely divisible (see, e.g., Levy [1948] or Donati-Martin, Doumerc,Matsumoto, and Yor [2004, Section 2.C] and also Remark 3.5.2). It turns outthat affine processes are infinitely decomposable Markov processes if and onlyif α = 0 or the symmetric cone corresponds to R+ or to the two-dimensionalLorentz cone. This is stated in Theorem 3.6.6 below.

We first prove some technical lemmas:

Lemma 3.6.2. Let g : K → R be an additive function, that is, g satisfiesCauchy’s functional equation

g(x+ y) = g(x) + g(y), x, y ∈ K. (3.65)

Then g can be extended to an additive function f : V → R. Moreover, if g ismeasurable on K then f is measurable on V . In that case, f is a continuouslinear functional, that is, f(x) = 〈c, x〉 for some c ∈ V .

Proof. The first part follows from the fact that K is generating.Concerning measurability, let E ∈ B(R) be a Borel measurable set. Then

we have by the additivity of f

f−1(E) =∞⋃n=1

Bn =∞⋃n=1

x+ ne | x ∈ V, f(x) ∈ E, ‖x‖ ≤ n − ne

=∞⋃n=1

y ∈ V | f(y) ∈ E + f(ne), ‖y − ne‖ ≤ n − ne

=∞⋃n=1

y ∈ K | g(y) ∈ E + g(ne), ‖y − ne‖ ≤ n − ne,

which is, in view of the measurability of g on K, again a measurable set.For x ∈ V we write x = (xi)i, where 1 ≤ i ≤ n. We introduce the

additive functions fi : R → R via fi(xi) = f(0, . . . , 0, xi, 0, . . . , 0). By themeasurability of f , we infer that all fi are measurable functions on R. By Aczeland Dhombres [1989, Chapter 2, Theorem 8], any additive measurable functionon the real line is a continuous linear functional. Hence, for each i, we inferthe existence of ci ∈ R such that fi(xi) = cixi holds. Since f(x) =

∑i fi(xi),

it follows that f(x) = 〈c, x〉 for some c ∈ V .


Let us now consider Cauchy’s exponential equation for h : K → R+, thatis,

h(x+ y) = h(x)h(y), x, y ∈ K. (3.66)

Lemma 3.6.3. Suppose h : K → R+ is measurable, strictly positive, andsatisfies (3.66). Then h(x) = e−〈c,x〉 for some c ∈ V . If h ≤ 1, then c ∈ K.

Proof. Since h is strictly positive, its logarithm yields the well-defined functiong : K → R, g(x) := log h(x). Clearly g is additive, hence by the first partof Lemma 3.6.2, there exists a unique additive extension f : V → R. Also,f is measurable on K, hence by the second assertion of Lemma 3.6.2 wehave f(x) = −〈c, x〉 for some c ∈ V . The last statement follows from themonotonicity of the exponential and the self duality of K.

Remark 3.6.4. The assumption of strict positivity of h in the precedinglemma is essential. Otherwise, there exist solutions h which are not of theasserted form.

Lemma 3.6.3 is the main ingredient of the proof of the following charac-terization concerning k-fold convolutions of Markov processes:

Lemma 3.6.5. Let (P(i)x )x∈K ∈ P (i = 0, 1, . . . , k). Then

P(1)

x(1) ∗ · · · ∗ P(k)

x(k) = P(0)x , ∀x(i) ∈ K, x = x(1) + · · ·+ x(k), (3.67)

if and only if, for all t = (t1, . . . , tN) ∈ RN+ and u = (u(1), . . . , u(N)) ∈

(K)N , N ∈ N, there exists 0 < ρ(i)(t,u) ≤ 1 and ψ(t,u) ∈ K such that∏ki=1 ρ

(i)(t,u) = ρ(0)(t,u) and

E(j)x

[e−

∑Ni=1〈u(i),Xti 〉

]= ρ(j)(t,u)e−〈ψ(t,u),x〉, ∀x ∈ K, j = 0, 1, . . . , k.

(3.68)

Proof. We proceed similarly as in the proof of Duffie et al. [2003, Lemma10.3]. Fix k > 1, N > 1, t, u and set

g(j)(x) := E(j)x

[e−

∑Ni=1〈u(i),Xti 〉

].

By the definition of the convolution, (3.67) is equivalent to the following

g(1)(x(1)) · . . . · g(k)(x(k)) = g(0)(x), ∀x(i) ∈ K, x = x(1) + · · ·+ x(k). (3.69)

Hence the implication (3.68) ⇒ (3.69) is obvious. For the converse directionwe observe that g(i) are strictly positive on all of K. Thus by (3.69) we have

g := g(1)/g(1)(0) = · · · = g(k)/g(k)(0) = g(0)/g(0)(0)

3.6. Relation to Infinitely Divisible Distributions 109

and g is a measurable, strictly positive function on K satisfying (3.66). Hencean application of Lemma 3.6.3 yields the validity of equation (3.68), whereρ(i)(t,u) = g(i)(0). By the definition of g(i), it follows that 0 < ρ(i)(t,u) ≤ 1and ψ(t,u) ∈ K.

Theorem 3.6.6. Let (Px)x∈K ∈ P and suppose that the rank and the Peirceinvariant of the Euclidean Jordan algebra V satisfy r > 1 and d > 0. Thenthe following assertions are equivalent:

(i) (Px)x∈K is infinitely decomposable.

(ii) (X, (Px)x∈K) is affine with vanishing diffusion parameter α = 0.

(iii) (X, (Px)x∈K) is affine and infinitely divisible.

Proof. (i)⇒(ii): Due to Lemma 3.6.5, infinite decomposability implies that Xis affine. Also, by the definition of infinite decomposability and by Lemma 3.6.5we have that the kth root (P(k)

x ) for each k ≥ 1 is an affine process with state

space K and exponents ψ(t, u) and φ(t, u)/k. This implies that (P(k)x )x∈K

has admissible parameters (α, b/k,B, c/k, γ,m/k, µ). Hence the admissibilitycondition proved in Proposition 3.2.6 implies b/k d(r− 1)α 0, for each k,which is impossible, unless α = 0.

(ii)⇒(iii): This implication follows from Proposition 2.5.2, in view of theLevy-Khintchine form of −φ(t, ·)− 〈ψ(t, ·), x〉, for each t > 0.

(iii)⇒(i): By assumption, every transition kernel pt(x, dξ) of X is infinitelydivisible with Laplace transform Pte

−〈u,x〉 = e−φ(t,u)−〈x,ψ(t,u)〉. For each k ≥ 1,the maps φ(k) := φ

k, ψ(k) := ψ satisfy the properties (2.3)–(2.4). Moreover,

infinite divisibility implies that for each (t, x) ∈ R+ ×K

Q(k)t e−〈u,x〉 := e−φ

(k)(t,u)−〈ψ(k)(t,u),xk〉

is the Laplace transform of a sub-stochastic measure on K. Together withthe properties (2.3)–(2.4) we may thus conclude that Q

(k)t induces a Feller

semigroup on C0(K), which is affine in y = x/k. Hence, for each k ≥ 1, wehave constructed a kth root of X, which is by the definition of its character-istic exponents φ(k), ψ(k) stochastically continuous. Whence Theorem 3.6.6 isproved.

Remark 3.6.7. A consequence of the above theorem is that any affine processon R+ or on the two-dimensional Lorentz cone is infinitely divisible. Thisalso follows from Duffie et al. [2003, Theorem 2.15]. For all other irreduciblesymmetric cones of dimension greater than 2, the boundary is curved implyingthe condition b d(r − 1)α such that the marginal distributions of an affineprocess can only be infinitely divisible if α = 0.


3.7 Results for Positive Semidefinite Matrices

For reasons of practical relevance and potential applications in mathematicalfinance, we reformulate and summarize in the following theorem the previ-ous results in the specific context of positive semidefinite matrices. Theseresults are also published in Cuchiero et al. [2011a] and were slightly refinedin Mayerhofer [2011].

Theorem 3.7.1. Let X be an affine process on S+r with r > 1. Then X is

regular and has the Feller property. Moreover, φ and ψ satisfy the generalizedRiccati equations for u ∈ S+

r , that is,

∂φ(t, u)

∂t= F (ψ(t, u)), φ(0, u) = 0, (3.70)

∂ψ(t, u)

∂t= R(ψ(t, u)), ψ(0, u) = u, (3.71)

and there exists an admissible parameter set (α, b, B, c, γ,m, µ) associated withthe truncation function χ = 0 such that the functions F and R are of thefollowing form

F (u) = 〈b, u〉+ c−∫S+r

(e−〈u,ξ〉 − 1)m(dξ),

R(u) = −2uαu+B>(u) + γ −∫S+r

(e−〈u,ξ〉 − 1

)µ(dξ).

Conversely, let (α, b, B, c, γ,m, µ) be an admissible parameter set. Thenthere exists a unique affine process on S+

r such that (2.1) holds for all (t, u) ∈R+ × S+

r , where φ(t, u) and ψ(t, u) are given by (3.70) and (3.71).

Proof. The first assertion is just a reformulation of Theorem 3.3.3 and anapplication of Proposition 3.2.3. The second one follows from Theorem 3.4.13.

The following theorem suggests that affine processes on S+r can be seen as

solutions of some (generalized) stochastic differential equations with jumps.

Theorem 3.7.2. Let X be a conservative affine process on S+r with r > 1

and let (α, b, B, c = 0, γ = 0,m, µ) be the related admissible parameter setassociated with the truncation function χ = 0. Then X is a semimartingale,

3.7. Results for Positive Semidefinite Matrices 111

whose characteristics (B,C, ν) with respect to χ = 0 are given by

Ct,ijkl =

∫ t

0

Cijkl(Xs)ds, (3.72)

Bt =

∫ t

0

(b+B(Xs)) ds, (3.73)

ν(dt, dξ) = (m(dξ) +M(Xt, dξ)) dt, (3.74)

where Cijkl(x) satisfies

Cijkl(x) = xikαjl + xilαjk + xjkαil + xjlαik, (3.75)

and M(x, dξ) is defined in (3.29). Furthermore, there exists, possibly on anenlargement of the probability space, a r × r-matrix of standard Brownianmotions W such that X admits the following representation

Xt = x+Bt +

∫ t

0

(√XsdWsΣ + Σ>dWs

√Xs

)+

∫ t

0

∫S+r

ξ µX(ds, dξ),

(3.76)

where Σ is an r × r matrix such that Σ>Σ = α and µX denotes the randommeasure associated with the jumps of X.

Moreover, let X ′ be a solution of (3.76) defined on some filtered probabilityspace (Ω′,F ′, (F ′t),P′) with P′[X0 = x]. Then P′ X ′−1 = Px.

Remark 3.7.3. By the above theorems it is easily seen that the so-calledWishart processes, which are the unique solutions (in law) of the followingstochastic differential equation

dXt = (b+MXt +XtM>)dt+

√XtdWtΣ + Σ>dW>

t

√Xt,

are particular affine processes on the cone of positive semidefinite matrices.

Proof. It follows from Theorem 1.4.8 that X is a semimartingale with charac-teristics (3.72)–(3.74). The canonical semimartingale representation (see Ja-cod and Shiryaev [2003, Theorem II.2.34]) of X is thus given by

Xt = x+Bt +Xct +

∫ t

0

∫S+r

ξµX(ds, dξ),

where Xc denotes the continuous martingale part and µX the random measureassociated with the jumps of X. In order to establish representation (3.76),we find it convenient to consider the vectorization, vec(Xc) ∈ Rr2

, of Xc.


The aim is now to find an r2-dimensional Brownian motion W on a possiblyenlarged probability space and an r2 × r2-matrix-valued function σ such that

vec(Xct ) =

∫ t

0

σ(Xs)dWs. (3.77)

Thus σ has to fulfill

d〈Xcij, X

ckl〉t = Xt,ikαjl +Xt,ilαjk +Xt,jkαil +Xt,jlαik = (σ(Xt)σ

>(Xt))ijkl.

(3.78)

Defining the entries of the r2×r2-matrix σ(x) by σijkl(x) =√xikΣlj+Σ>il

√xjk,

yields Cijkl(x) = (σ(x)σ>(x))ijkl. Hence σ(x) satisfies (3.78). Analogous tothe proof of Rogers and Williams [1987, Theorem V.20.1], we can now build

an r2-dimensional Brownian motion W on an enlargement of the probabilityspace such that (3.77) holds true. As the (ij)th entry of Xc is given by

Xct,ij = vec(Xc

t )ij =

∫ t

0

d∑k,l=1

σijkl(Xs)dWs,kl

=

∫ t

0

(√XsdWsΣ + Σ>dW>

s

√Xs

)ij,

where W is the r × r-matrix Brownian motion satisfying vec(W ) = W , weobtain the desired representation.

Concerning the second part of the theorem, let A denote the integro-differential operator defined on C2

b (S+r ) by

Af(x) =1

2

∑i,j,k,l

Cijkl(x)∂2f(x)

∂xij∂xkl+∑i,j

(bij +Bij(x))∂f(x)

∂xij

+

∫S+r

(f(x+ ξ)− f(x)) (m(dξ) +M(x, dξ)).

Then our assumption implies that P′ is a solution of the martingale problemfor A, meaning that

f(X ′t)− f(X ′0)−∫ t

0

Af(X ′s−)ds

is a martingale for all f ∈ C2c (S+

r ). Since Px is the unique solution of themartingale problem on (Ω,F) (see Cuchiero et al. [2011a, Proof of Proposition5.9]), we thus have P′ X ′−1 = Px.

Part II

Polynomial Processes

113

Chapter 4

Characterization and Relationto Semimartingales

Similar to affine processes, we define polynomial processes as a particular classof time-homogeneous Markov processes with state space S ⊆ Rn (augmentedby a point ∆ /∈ S), where S denotes some closed subset of Rn. We use thesame setting as in Chapter 1, with the only difference that we assume a priorithat the time-homogeneous (not necessarily conservative) Markov processesX defined on a filtered space (Ω,F , (Ft)t≥0) has cadlag paths and that thefiltration (Ft) is right-continuous.

Recall that we considered affine processes on the space (Ω,F , (Ft)t≥0,Px),where Ω is the space of cadlag paths defined in Remark 1.2.12 and F ,Ftare given in (1.42). Since (Ω,F , (Ft)t≥0,Px) does not satisfy the usual con-ditions, we here do not assume them either and require the filtration onlyto be right-continuous such that affine processes under moment conditionscan be regarded as a subclass of polynomial processes (see Definition 4.1.1below). This is in line with the setting of Jacod and Shiryaev [2003], wherethe stochastic basis is not assumed to be complete either. As the selection ofversions with more regular trajectories might be a delicate issue, we alwaysassume that semimartingales have cadlag trajectories.

Moreover, in contrast to Chapter 1, we do not restrict the definition of theMarkov property to bounded functions, but rather assume that

Ex[f(Xt+s)|Fs] = EXs [f(Xt)], Px-a.s.

holds for all x ∈ S∆, s, t ∈ [0,∞) and all Borel functions f : S∆ → R satisfyingEx[|f(Xt)|] <∞ for all t ≥ 0 and x ∈ S. Similarly, the semigroup (Pt)t≥0,

Ptf(x) := Ex[f(Xt)] =

∫S

f(ξ)pt(x, dξ), x ∈ S∆,

115

116 Chapter 4. Characterization and Relation to Semimartingales

is also defined for all Borel measurable functions f : S∆ → R satisfyingEx[|f(Xt)|] <∞ for all t ≥ 0 and x ∈ S.

4.1 Definition and Characterization

We denote by Pol≤m(S) the finite dimensional vector space of polynomials upto degree m ∈ N on S, i.e., the restriction of polynomials on Rn to S, definedby

Pol≤m(S) :=

S 3 x 7→m∑|k|=0

αkxk, ∆ 7→ 0

∣∣∣αk ∈ R

,

where we use multi-index notation k = (k1, . . . , kn) ∈ Nn, |k| = k1 + · · · + knand xk = xk1

1 · · ·xknn . The dimension of Pol≤m(S) is denoted by N < ∞ anddepends on the state space S.

Definition 4.1.1. We call an S∆-valued time-homogeneous Markov processm-polynomial if

(i) for all 0 ≤ k ≤ m, all f ∈ Pol≤k(S), x ∈ S and t ≥ 0,

x 7→ Ptf(x) ∈ Pol≤k(S),

(ii) t 7→ Ptf(x) is continuous at t = 0 for all f ∈ Pol≤m(S).

If X is m-polynomial for all m ≥ 0, then it is called polynomial.

Remark 4.1.2. (i) From the above definition it follows that

Pt|f |(x) = Ex[|f(Xt)|] <∞

for every f ∈ Pol≤m(S), x ∈ S and t ≥ 0. In other words, an m-polynomial process always satisfies Ex[‖Xt‖m] < ∞ for all x ∈ S andt ≥ 0.

(ii) The subtlety of Definition 4.1.1 lies in the fact that we assume

Pt Pol≤k(S) ⊂ Pol≤k(S)

for all 0 ≤ k ≤ m (compare with Remark 4.1.10 (iv)). The assumptionthat Pt Pol≤m(S) ⊂ Pol≤m(S) only for m, but not for smaller degrees isnot sufficient for our proof of the equivalence of (iv) in Theorem 4.1.8to the other assertions. The equivalence of (i), (ii), (iii) can however beproved by the same arguments.

4.1. Definition and Characterization 117

(iii) Throughout this chapter we shall always use continuity assumptions ofthe type “ t 7→ Ptf(x) is continuous for every x ∈ S ”, which are easierto verify than strong continuity with respect to some chosen norm. Forf ∈ Pol≤m(S), the existence of moments of degree m+ ε for some ε > 0is for examples already sufficient.

Let us introduce the following two notions of Markov generators, which weshall use to characterize m-polynomial processes.

Definition 4.1.3. An operator A with domain DA is called infinitesimal gen-erator for X if DA consists of those functions f : S∆ → R which satisfyPt|f |(x) < ∞ for all t ≥ 0 and x ∈ S and for which there exists a functionAf such that the process

f(Xt)− f(X0)−∫ t

0

Af(Xs)ds, (4.1)

is a (Ft,Px)-martingale for every x ∈ S∆.

Remark 4.1.4. If f lies in the domain of the infinitesimal generator, thendue to the martingale property all increments of f(Xt)−f(X0)−

∫ t0Af(Xs)ds

have vanishing expectation, i.e., for all s < t,

Ex[f(Xt)− f(Xs)−

∫ t

s

Af(Xu)du

]= 0.

In particular, by Fubini’s theorem,∫ t

0PuAf(x)du exists on finite time intervals

and thus also Pu|Af |(x) for almost all u with respect to the Lebesgue measure.

The following notion of the extended infinitesimal generator is due toDynkin (see, e.g., Cinlar et al. [1980, Definition 7.1]).

Definition 4.1.5. An operator G with domain DG is called extended in-finitesimal generator for X if DG consists of those Borel measurable functionsf : S∆ → R for which there exists a function Gf such that the process

f(Xt)− f(X0)−∫ t

0

Gf(Xs)ds (4.2)

is well-defined and a (Ft,Px)-local martingale for every x ∈ S∆.

Remark 4.1.6. As in Chapter 1, we define the lifetime of the process by

T∆(ω) = inft |Xt(ω) = ∆, (4.3)


where the infimum over the empty set is set to be ∞. Since T∆ < t =⋃q<t,q∈QXq = ∆ ∈ Ft and as (Ft) is supposed to be right-continuous, T∆ is

an Ft-stopping time. Due to our convention f(∆) = 0, the local martingaleproperty of (4.2) is therefore equivalent to

f(Xt)1t<T∆ − f(X0)−∫ t

0

Gf(Xs)1s<T∆ds

being a local martingale. The analogous statement holds true for (4.1).

Since our definition of the infinitesimal generator in (4.1) is not the stan-dard one used in the literature for Feller processes (compare Revuz and Yor[1999, Chapter VII]), we formulate the following lemma.

Lemma 4.1.7. If f lies in the domain of the infinitesimal generator, f ∈ DA,then we have:

(i) Ptf ∈ DA and APtf = PtAf for every t ≥ 0.

(ii) If t 7→ PtAf(x) is continuous at t = 0, then Ptf is solves the Kol-mogorov backward equation in the strong sense, i.e.,

∂u(t, x)

∂t= Au(t, x), u(0, x) = f(x).

Proof. For the first statement, we show that

Ptf(Xh)− Ptf(X0)−∫ h

0

PtAf(Xs)ds

is a (Fh,Px)-martingale for any fixed t ≥ 0. By the definition of the infinites-imal generator, this then implies that Ptf ∈ DA and APtf = PtAf . Indeed,since f lies in DA, we have by Definition 4.1.3 and Remark 4.1.4 that f(Xs)and Af(Xs) are integrable for every s ≥ 0, hence Ptf(Xs) and PtAf(Xs) aswell. Therefore the following expectation is well-defined and we obtain foru ≤ h

Ex[Ptf(Xh)− Ptf(X0)−

∫ h

0

PtAf(Xs)ds∣∣∣Fu]

= Ptf(Xu)− Ptf(x)−∫ u

0

PtAf(Xs)ds

+ Ex[Ptf(Xh)− Ptf(Xu)−

∫ h

u

PtAf(Xs)ds∣∣∣Fu] .


By the Markov property, the conditional expectation on the right is equal to

EXu[Ptf(Xh−u)− Ptf(X0)−

∫ h−u

0

PtAf(Xs)ds

].

But for any y ∈ S∆, we have

Ey[Ptf(Xh−u)− Ptf(X0)−

∫ h−u

0

PtAf(Xs)ds

]= Pt+h−uf(y)− Ptf(y)−

∫ t+h−u

t

PsAf(y)ds

= 0,

where the last equality follows from Remark 4.1.4. This completes the proofof (i).

Statement (ii) follows from the continuity of t 7→ PtAf(x) and from asser-tion (i), since

∂Ptf(x)

∂t= lim

h→0

Pt+hf(x)− Ptf(x)

h= lim

h→0PtPhf(x)− f(x)

h

= limh→0

Pt1

h

∫ h

0

PsAf(x)ds = PtAf(x) = APtf(x).

Let us now state our main theorem:

Theorem 4.1.8. Let X be a time-homogeneous Markov process with statespace S∆ and semigroup (Pt). Additionally we assume that t 7→ Ptf(x) iswell-defined and continuous at t = 0 for all f ∈ Pol≤m(S). Then the followingassertions are equivalent:

(i) X is m-polynomial for some m ≥ 0.

(ii) For every 0 ≤ k ≤ m, there exists a linear map A : Pol≤k(S) →Pol≤k(S), such that, for all t ≥ 0, (Pt) restricted to Pol≤k(S) can bewritten as

Pt|Pol≤k(S) = etA.

(iii) Pol≤m(S) lies in the domain of the infinitesimal generator, i.e., for allf ∈ Pol≤m(S), x ∈ S∆ and t ≥ 0,

f(Xt)− f(X0)−∫ t

0

Af(Xs)ds

is a (Ft,Px)-martingale, and A(Pol≤k(S)) ⊂ Pol≤k(S) for all 0 ≤ k ≤m.


Moreover, the following weaker statement is obviously implied by (i), (ii) and(iii), but is also equivalent if m ≥ 2 is an even number.

(iv) Pol≤m(S) lies in the domain of the extended infinitesimal generator, i.e.,for all f ∈ Pol≤m(S), x ∈ S∆ and t ≥ 0,

M ft := f(Xt)− f(X0)−

∫ t

0

Gf(Xs)ds

is a (Ft,Px)-local martingale and G(Pol≤k(S)) ⊂ Pol≤k(S) for all 0 ≤k ≤ m.

Remark 4.1.9. From the point of view of applications the most importantimplication is (iv) ⇒ (ii) (see Chapter 6).

Proof. Our strategy to prove the above equivalences is to show (i) ⇒ (ii) ⇒(iii) ⇒ (i) and (iv) ⇒ (iii) under the additional assumption that m ≥ 2 isan even number. In order to prove the last implication, we need some resultsderived in Proposition 4.2.1 and Lemma 4.2.3 below. Therefore we postponethe proof of (iv) ⇒ (iii) to Subsection 4.3.1.

Throughout the proof, let 0 ≤ k ≤ m be fixed. We start by showing (i)⇒ (ii). Let L(Pol≤k(S)) denote the space of all linear maps from Pol≤k(S) toPol≤k(S). By the semigroup property,

P(·)|Pol≤k(S) : R+ → L(Pol≤k(S)) (4.4)

satisfies the Cauchy functional equationPt+s = PtPs for all t, s ≥ 0,P0 = I.

Since Pol≤k(S) is finite dimensional, continuity of (4.4) at t = 0 already impliesthat there exists some A ∈ L(Pol≤k(S)) such that Pt|Pol≤k(S) = etA (see Engeland Nagel [2000, Theorem I.2.9]).

Next, we show (ii) ⇒ (iii). By (ii) we have, for every f ∈ Pol≤k(S),Af ∈ Pol≤k(S) and

Ptf − f −∫ t

0

PsAfds = etAf − f −∫ t

0

esAAfds = 0.

Thus f(Xt)−f(x)−∫ t

0Af(Xs)ds is a (Ft,Px)-martingale. Hence f lies in DA

and Af = Af , implying that A(Pol≤k(S)) ⊂ Pol≤k(S) holds true.


In order to prove (iii) ⇒ (i), we consider the Kolmogorov backward equa-tion (in the strong sense) for an initial value u(0, ·) = f ∈ Pol≤k(S):

∂u(t, x)

∂t= Au(t, x).

By Lemma 4.1.7 (ii), Ptf solves the Kolmogorov equation (in the strong sense),since t 7→ PtAf(x) is continuous at t = 0 for any f ∈ Pol≤k(S). This followsfrom the fact that A maps Pol≤k(S) to itself and the continuity assumptionon t 7→ Ptf(x) for any f ∈ Pol≤k(S). By choosing a basis 〈e1, . . . , eN〉 ofPol≤k(S), we can define a linear map A on Pol≤k(S) by setting

Aek =:N∑l=1

Aklel.

Then A|Pol≤k(S) = A and the Kolmogorov backward equation can be under-stood as a linear ODE in the classical sense, whose unique solution is givenby etAf . The map Ps(e

(t−s)Af) is therefore constant with respect to s, sinceits first derivative vanishes, that is,

d

dsPse

(t−s)Af(x) = PsAe(t−s)Af(x)− PsAe(t−s)Af(x) = 0,

where we apply Lemma 4.1.7 (i). Hence, on Pol≤k(S), Ptf is equal to etAfand is therefore a polynomial of degree smaller than or equal to k. Since thisholds true for any 0 ≤ k ≤ m, X is m-polynomial.

Remark 4.1.10. (i) There is no need in Theorem 4.1.8 (ii) to restrict thetime parameter t to R+, since, for t ∈ R, (etA) extends to a group.

(ii) If X is an m-polynomial process, then the process Z = (X,X2, . . . , Xm)is a 1-polynomial process. If m is even, the analysis of m-polynomialprocesses could be reduced to the study of 2-polynomial processes at thecost of a more complicated state space, which is due to the constructionZ ′ = (X,X2, . . . , X

m2 ).

(iii) Let us remark that the implication (iv) ⇒ (iii) in Theorem 4.1.8 onlyholds true if m ≥ 2. In the case m = 1, the inverse 3-dimensionalBessel process X is an example showing that the extended infinitesimalgenerator maps Pol≤1(R+) to Pol0(R+), while

Xt −X0 −∫ t

0

GXsds = Xt −X0 (4.5)


is a strict local martingale (see Protter [2005, Example I.6.2]). Indeed,the inverse 3-dimensional Bessel process defined by X = 1

‖B‖ , where Bdenotes a 3-dimensional Brownian motion started at B0 6= 0, satisfies

dXt = −X2t dWt, X0 =

1

‖B0‖,

where W is a one-dimensional standard Brownian motion. The extendedinfinitesimal generator is therefore given by

Gf(x) =1

2x4d

2f(x)

dx2.

Hence G(Pol≤1(R+)) = 0. However, since (4.5) fails to be a true mar-tingale, X is not a 1-polynomial process.

(iv) In Definition 4.1.1 we require m-polynomial processes to be also k-poly-nomial for all 0 ≤ k ≤ m, that is, we implicitly exclude processes whoseextended infinitesimal generator maps polynomials of degree k < mto polynomials of degree greater than k ≤ m, while G(Pol≤m(S)) ⊂Pol≤m(S) still holds true. The following process is an example of thistype. Consider

dXt =

(1

2− bXt +

1

2X2t

)dt+

√X2t (1−Xt)dWt, X0 = x ∈ [0, 1],

where b ≥ 1 and W is a one-dimensional standard Brownian motion.In this case, the state space is the interval S = [0, 1]. Existence of weaksolutions for this SDE follow from the continuity of the drift and thediffusion component (see Rogers and Williams [1987, Theorem V.23.5]).Moreover, we have

Gf(x) =

(1

2− bx+

1

2x2

)df(x)

dx+

1

2x2(1− x)

d2f(x)

dx2.

Thus G(Pol≤1(S)) ⊂ Pol≤2(S), while G(Pol≤2(S)) ⊂ Pol≤2(S). Usingthe arguments of Lemma 4.2.3 below, it follows that

f(Xt)− f(X0)−∫ t

0

Gf(Xs)ds

is a true martingale for f ∈ Pol≤2(S) due to compactness of the statespace. By the same arguments as in the proof of Theorem 4.1.8 (iii)⇒ (i), it then follows that Pt(Pol≤2(S)) ⊂ Pol≤2(S) but Pt(Pol≤1(S)) *Pol≤1(S).

4.2. Polynomial Semimartingales 123

The following corollary is a basic conclusion of Theorem 4.1.8 (ii) andestablishes a link to time-space harmonic functions (see, e.g., Sole and Utzet[2008]).

Corollary 4.1.11. Let X be an m-polynomial process with semigroup (Pt) andlet f ∈ Pol≤m(S) be fixed. Then there exists a unique function Q : R×S∆ → R,being real analytic in time and satisfying Q(t, ·) ∈ Pol≤m(S) for all t ∈ R suchthat Q(0, x) = f(x) and Q(t− s,Xs) is a (Fs,Px)-martingale for s ≥ 0.

Proof. We prove that Q(t, x) is given by Ptf(x). As X is m-polynomial,Ptf ∈ Pol≤m(S) and by Theorem 4.1.8 (ii) and Remark 4.1.10 (i) it is areal analytic function in time. Clearly, P0f(x) = f(x) and by the Markovproperty Pt−sf(Xs) is a (Fs,Px)-martingale. Concerning uniqueness, let nowQ : R × S∆ → R be a function satisfying Q(0, x) = f(x) and the martingaleproperty. The latter implies for all s ≥ 0

Q(t, x) = Ex[Q(t− s,Xs)].

Thus, by setting s = t, we obtain

Q(t, x) = Ex[Q(0, Xt)] = Ex[f(Xt)] = Ptf(x),

as Q(0, Xt) = f(Xt). This proves uniqueness.

Remark 4.1.12. For t = 0, Q(−s, x) as defined in Corollary 4.1.11 can beconsidered as a time-space harmonic function (see, e.g., Sole and Utzet [2008])for the m-polynomial process X.

4.2 Polynomial Semimartingales

In our setting defined above we neither assumed the Markov process to be con-servative, nor specified the domain of the (extended) infinitesimal generator,for example by requiring the family of functions ei〈u,·〉 |u ∈ Rn to lie in DA orDG. Therefore the process X with X0 = x is in general not a semimartingalewith respect to the stochastic basis (Ω,F , (Ft),Px). However, if a Markovprocess satisfies the assumption of Theorem 4.1.8 (iv), then (Xt1t<T∆) isa special semimartingale whose characteristics are absolutely continuous withrespect to the Lebesgue measure, as shown in Proposition 4.2.1 below. This re-markable property follows from the assumption that G(Pol≤m(S)) ⊂ Pol≤m(S)and compensates for the fact that Pol≤m(S) is not a complete class in the ter-minology of Cinlar et al. [1980, Definition 7.8]. Indeed, according to Cinlaret al. [1980, Theorem 7.16] a Markov process is a semimartingale with ab-solutely continuous characteristics if and only if the domain of its extendedgenerator is a full and complete class.


Proposition 4.2.1. Let X be a time-homogeneous Markov process with statespace S∆. Let m ≥ 2 and assume that Pol≤m(S) lies in the domain of theextended infinitesimal generator and that G maps Pol≤k(S) to itself for all0 ≤ k ≤ m. Then (Xt1t<T∆) is a special semimartingale with respect tothe stochastic basis (Ω,F , (Ft),Px). The components of its characteristics(B,C, ν) associated with the “truncation function” χ(ξ) = ξ satisfy1

Bt,i =

∫ t

0

(GXs,i)1s<T∆ds =:

∫ t

0

bi(Xs)1s<T∆ds, (4.6)

Ct,ij +

∫ t

0

∫Rnξiξjν(ds, dξ) =:

∫ t

0

aij(Xs)1s<T∆ds, (4.7)

where bi ∈ Pol≤1(S) and aij ∈ Pol≤2(S). Moreover, the characteristics C andν can be written as

Ct,ij =

∫ t

0

cs,ijds, ν(ω; dt, dξ) = Kω,t(dξ)dt, (4.8)

where (cij)i,j≤n is a predictable process and Kω,t(dξ) is a predictable randommeasure on (Rn,B(Rn)). Finally, we have for all 3 ≤ |k| ≤ m

∫RnξkKω,t(dξ) =

|k|∑|l|=0

αlXlt(ω)1t<T∆(ω), for almost all t ≥ 0, (4.9)

with some coefficients αl and for all 2 ≤ k ≤ 2bm2c∫

Rn‖ξ‖kKω,t(dξ) ≤ C

(1 + ‖Xt(ω)1t<T∆(ω)‖2b k+1

2c), (4.10)

for almost all t ≥ 0 and some constant C.

Remark 4.2.2. (i) Due to the fact that Gf(x) is a well-defined polynomialfor every f ∈ Pol≤m(S), it follows that

∫Rn ξ

kKω,t(dξ) exists for all2 ≤ |k| ≤ m. This means in particular that

∫Rn ‖ξ‖

kKω,t(dξ) < ∞ forall 2 ≤ k ≤ m.

(ii) We shall make use of the very condensed notation GXs,i := G pri(Xs),where we consider the projection on the i-th component. Analogously weapply this to quadratic monomials.

1All statements concerning the characteristics are meant up to an evanescent set.


(iii) The above proposition asserts that a Markov process which satisfies theassumption of Theorem 4.1.8 (iv) is a semimartingale with absolutelycontinuous characteristics. In particular, the drift part is an affine func-tion in X, the modified second characteristic, which corresponds to

Ct,ij +

∫ t

0

∫Rnξiξjν(ds, dξ)

(see Jacod and Shiryaev [2003, Definition II.2.16, Proposition II.2.17]),is a quadratic polynomial, and the moments of the compensator of thejump measure are polynomials in X of the same degree.

Proof. For all f ∈ Pol≤m(S)

M ft = f(Xt)1t<T∆ − f(X0)−

∫ t

0

(Gf(Xs))1s<T∆ds, (4.11)

is a (Ft,Px)-local martingale. Here, T∆ denotes the lifetime defined in (4.3).As the process

∫ t0(Gf(Xs))1s<T∆ds is predictable, (f(Xt)1t<T∆) is a special

R-valued semimartingale for all f ∈ Pol≤m(S). In particular, f(Xt)1t<T∆has cadlag paths implying that |f(XT∆−)| < ∞ (notice here that f(∆) =0) and f(X) cannot explode. Choosing f(x) = xi for i = 1, . . . , n, thenimplies that (Xt1t<T∆) is an (n-dimensional) special semimartingale. Letnow (B,C, ν) denote the characteristics of (Xt1t<T∆) with respect to the“truncation function” χ(ξ) = ξ. Since

Xt,i1t<T∆ = X0,i +Mxit +

∫ t

0

(GXs,i)1s<T∆ds,

it follows by the unique decomposition of a special semimartingale into a localmartingale and a predictable finite variation process that

Bt =

∫ t

0

(GXs,i)1s<T∆ds.

Setting bi(x) := Gxi (notice here Remark 4.2.2 (ii)) implies that bi ∈ Pol≤1(S).In order to determine the properties of C and ν, let us consider the poly-

nomials f(x) = xixj for 1 ≤ i, j ≤ n. Then the special semimartingaleXt,iXt,j1t<T∆ can be written as

Xt,iXt,j1t<T∆ = X0,iX0,j +Mxixjt +

∫ t

0

(GXs,iXs,j)1s<T∆ds. (4.12)


By Ito’s formula, we have on the other hand

Xt,iXt,j1t<T∆ = X0,iX0,j +

∫ t

0

Xs−,i1s<T∆dMxjs

+

∫ t

0

Xs−,j1s<T∆dMxis +

∫ t

0

Xs,i1s<T∆bj(Xs)ds

+

∫ t

0

Xs,j1s<T∆bi(Xs)ds+ Ct,ij

+

∫ t

0

∫Rnξiξjµ

X1t<T∆(ds, dξ),

(4.13)

where µX1t<T∆ denotes the random measure associated with the jumps ofX1t<T∆. Since Mxi is a local martingale and Xs−,j is caglad,∫ t

0

Xs−,j1s<T∆dMxis

is also a local martingale (see, e.g., Protter [2005, Theorem III.7.33]) for alli, j = 1, . . . , n. Furthermore,∫ t

0

Xs,i1s<T∆bj(Xs)ds+

∫ t

0

Xs,j1s<T∆bi(Xs)ds+ Ct,ij

is a predictable process of finite variation, thus in particular a process oflocally integrable variation by Jacod and Shiryaev [2003, Lemma I.3.10]. AsXt,iXt,j1t<T∆ is a special semimartingale, it follows from Jacod and Shiryaev[2003, Proposition I.4.22] that∫ t

0

∫Rnξiξjµ

X1t<T∆(ds, dξ)

is also of locally integrable variation, since it is of finite variation and for aspecial semimartingale the finite variation part is locally integrable. Thereforewe have by Jacod and Shiryaev [2003, Proposition II.1.28] that∫ t

0

∫Rnξiξjµ

X1t<T∆(ds, dξ)−∫ t

0

∫Rnξiξjν(ds, dξ)

is a local martingale. Thus, putting (4.12) and (4.13) together, we find∫ t

0

(GXs,iXs,j)1s<T∆ds =

∫ t

0

Xs,ibj(Xs)1s<T∆ds

+

∫ t

0

Xs,jbi(Xs)1s<T∆ds+ Ct,ij

+

∫ t

0

∫Rnξiξjν(ds, dξ).


Since Gxixj and xibj(x) lie in Pol≤2(S), we have

Ct,ij +

∫ t

0

∫Rnξiξjν(ds, dξ) =

∫ t

0

aij(Xs)1s<T∆ds (4.14)

for some aij ∈ Pol≤2(S).

Let us now define A′t(ω) =∫ t

0

∫Rn ‖ξ‖

2ν(ω; ds, dξ). By the same argumentsas in Jacod and Shiryaev [2003, Proposition II.2.9.b], there exists a randommeasureK ′(ω, t; dξ) on (Rn,B(Rn)) such that ν(ω; dt, dξ) = K ′(ω, t; dξ)dAt(ω).Moreover, since

n∑i=1

Ct,ii(ω) + A′t(ω) =n∑i=1

∫ t

0

aii(Xs(ω))1s<T∆(ω)ds =:

∫ t

0

as(ω)ds

and as Ct,ii, i = 1, . . . , n, and A′t are nonnegative increasing processes (of finitevariation), Cii and A′ are absolutely continuous with respect to the Lebesguemeasure. Hence Jacod and Shiryaev [2003, Proposition I.3.13] implies theexistence of predictable processes cii andH such that Ct,ii =

∫ t0cs,iids andA′t =∫ t

0Hsds. Then Kω,t(dξ) = Ht(ω)K ′(ω, t; dξ) is again a predictable random

measure satisfying ν(ω; dt, dξ) = Kω,t(dξ)ds Px-a.s. Having constructed thiskernel, (4.14) now becomes

Ct,ij =

∫ t

0

(aij(Xs)1s<T∆ −

∫RnξiξjKω,t(dξ)

)ds,

implying that Ct,ij, for i 6= j, is also absolutely continuous with respect to the

Lebesgue measure and can therefore be written as Ct,ij =∫ t

0cs,ijds.

In order to establish properties (4.9) and (4.10), we consider the polynomial


f(x) = xk for 3 ≤ k = |k| ≤ m. As before, we obtain by Ito’s formula

f(Xt)1t<T∆ − f(X0)

=n∑i=1

∫ t

0

Dif(Xs−)1s<T∆dMxis +

n∑i=1

∫ t

0

Dif(Xs)1s<T∆bi(Xs)ds

+1

2

n∑i,j=1

∫ t

0

(Dijf(Xs)1s<T∆

(cs,ij +

∫RnξiξjKω,s(dξ)

))ds

+

∫ t

0

∫Rn

k∑|l|=3

(k

l

)Xk−ls 1s<T∆ξ

l

Kω,s(dξ)ds

+

∫ t

0

∫Rn

k∑|l|=2

(k

l

)Xk−ls− 1s<T∆ξ

l

µX1t<T∆(ds, dξ)

−∫ t

0

∫Rn

k∑|l|=2

(k

l

)Xk−ls 1s<T∆ξ

l

Kω,s(dξ)ds.

(4.15)

Since f(Xt)1t<T∆ is a special semimartingale, it follows by the same argu-ments as before that∫ t

0

∫Rn

k∑|l|=2

(k

l

)Xk−ls− 1s<T∆ξ

l

µX1t<T∆(ds, dξ)

is of locally integrable variation. This implies that the difference of the lasttwo terms in (4.15) is a local martingale. As Gf(x) = Gxk, Dif(x)b(x) andDijf(x)(cij +

∫Rn ξiξjKω,·(dξ)) lie in Pol≤k(S) for all k = |k| ≤ m, (4.9) then

follows from (4.11) and induction. Moreover, since 2bk+12c is even, we have

for 2 ≤ k ≤ 2bm2c and almost all t ≥ 0∫

Rn‖ξ‖kKω,t(dξ) ≤

∫Rn‖ξ‖2Kω,t(dξ) +

∫Rn‖ξ‖2b k+1

2cKω,t(dξ)

≤n∑i=1

aii(Xt)1t<T∆ +

2b k+12c∑

|l|=0

αlXlt1t<T∆

≤ C(

1 + ‖Xt1t<T∆‖2b k+12c),

where C denotes some constant.


Using the above derived semimartingale properties, we can now establisha maximal inequality, which we need for the proof of the implication (iv) ⇒(iii) in Theorem 4.1.8. A similar statement for the case of Levy driven SDEscan be found in Jacod, Kurtz, Meleard, and Protter [2005].

Lemma 4.2.3. Fix T > 0 and let m ≥ 2. Let (Xt1t<T∆) be a specialsemimartingale with state space S, whose characteristics (B,C, ν) associatedwith the “truncation function” χ(ξ) = ξ satisfy (4.6), (4.7) and (4.8). Then

there exists a constant C such that for all 0 ≤ t ≤ T

Ex[sups≤t

∥∥Xs1s<T∆∥∥m] ≤ C

(‖x‖m + 1 +

∫ t

0

Ex[‖Xs1s<T∆‖m

]ds

+

∫ t

0

Ex[∫

Rn‖ξ‖mKω,s(dξ)

]ds

).

(4.16)

Remark 4.2.4. Notice that this lemma remains also true when one of bothsides is infinite. The lemma should be considered as an assertion about specialsemimartigales with a particular structure on the characteristics.

Proof. Let Xt1t<T∆ = X0 + Mt + Bt be the canonical decomposition ofthe special semimartingale (Xt1t<T∆). By the following version of Jensen’sinequality ∣∣∣∣∣

k∑i=1

yi

∣∣∣∣∣m

≤ km−1

k∑i=1

|yi|m (4.17)

for m ≥ 1, we have

sups≤t

∣∣Xs,i1s<T∆∣∣m ≤ 3m−1

(|X0,i|m + sup

s≤t|Ms,i|m + sup

s≤t|Bs,i|m

).

Thus, in order to prove (4.16), it suffices to find estimates for sups≤t |Ms,i| and

sups≤t |Bs,i|. We remark that the constant C in the inequalities below mayvary from line to line. Since Bs,i =

∫ s0bi(Xu)1u<T∆du and bi ∈ Pol≤1(S), we

have

sups≤t

∣∣∣∣∫ s

0

bi(Xu)1u<T∆du

∣∣∣∣m ≤ (∫ t

0

|bi(Xu)| 1u<T∆du

)m≤∫ t

0

tm−1 |bi(Xu)|m 1u<T∆du

≤ C

(1 +

∫ t

0

∥∥Xu1u<T∆∥∥m du) .

(4.18)


Hence, by Fubini’s theorem,

Ex[sups≤t|Bs,i|m

]≤ C

(1 +

∫ t

0

Ex[‖Xs1t<T∆‖m

]ds

).

Concerning sups≤t |Ms,i|, an application of Burkholder-Davis-Gundy’s inequal-ity yields

Ex[sups≤t|Ms,i|m

]≤ CEx

[[Mi,Mi]

m2t

].

Since the finite variation process Bt,i is continuous, we have

[Mi,Mi]t = Ct,ii +

∫ t

0

∫Rnξ2i µ

X1t<T∆(ds, dξ) =: Ct,ii + Yt,

where µX1t<T∆ is the random measure associated with the jumps of X1t<T∆.

Applying again inequality (4.17), we obtain [Mi,Mi]m2t ≤ 2

m2−1(C

m2t,ii + Y

m2t ).

As C satisfies (4.7), we can estimate Ct,ii by Ct,ii ≤∫ t

0aii(Xs)1s<T∆ds, where

aii ≥ 0 ∈ Pol≤2(S). Using a similar reasoning as in (4.18), we get

Ex[C

m2t,ii

]≤ C

(1 +

∫ t

0

Ex[‖Xs1t<T∆‖m

]ds

).

Therefore it remains to handle Ym2t =

(∫ t0ξ2i µ

X1t<T∆(ds, dξ))m

2. Following

the approach of Jacod et al. [2005], let us define Tn = inft |Yt ≥ n. Then

Ym2t∧Tn =

∑s≤t∧Tn

(Ys− + ∆Ys)m2 − (Ys−)

m2

=

∫ t∧Tn

0

∫Rn

((Ys− + ξ2

i )m2 − (Ys−)

m2

)µX1t<T∆(ds, dξ),

which is due to the fact that Y is purely discontinuous, non-decreasing and∆Ys = |∆(Xs,i1s<T∆)|2. Furthermore, since ν is the predictable compensatorof µX1t<T∆ ,

Ex[Y

m2t∧Tn

]= Ex

[∫ t∧Tn

0

∫Rn

((Ys− + ξ2

i )m2 − (Ys−)

m2

)ν(ds, dξ)

], (4.19)

an equality which even remains true if one of both sides is infinite. In thesequel we shall use the following inequalities (see Jacod et al. [2005])

(y + z)p − yp ≤ 2p−1(yp−1z + zp), (4.20)

yp−1x ≤ εyp +xp

εp−1, (4.21)


for x, y, z ≥ 0, every ε > 0 and p ≥ 1. Applying (4.20), equation (4.19)becomes

Ex[Y

m2t∧Tn

]≤ Ex

[∫ t∧Tn

0

∫Rn

2m2−1(Y

m2−1

s− ξ2i + |ξi|m

)ν(ds, dξ)

].

For the first part, we then have due to the assumption on ν

Ex[∫ t∧Tn

0

∫Rn

2m2−1Y

m2−1

s− ξ2i ν(ds, dξ)

]= Ex

[∫ t∧Tn

0

2m2−1Y

m2−1

s

(∫Rnξ2iKω,s(dξ)

)ds

]≤ Ex

[∫ t∧Tn

0

2m2−1Y

m2−1

s aii(Xs)1s<T∆ds

],

where the last inequality follows from the fact that Ys is positive for all s ≤ t.Estimating aii(x) by C(1 + ‖x‖2), we obtain using (4.21),

Ex[Y

m2t∧Tn

]≤ Ex

[∫ t∧Tn

0

C

(εY

m2s +

1 + ‖Xs‖m

εm2−1

)1s<T∆ds

]+ Ex

[∫ t∧Tn

0

∫Rn

2m2−1|ξi|mν(ds, dξ)

]≤ CεEx

[nm2 ∧ Y

m2t∧Tn

]+ Ex

[∫ t∧Tn

0

C

εm2−1

(1 + ‖Xs‖m)1s<T∆ds

]

+ Ex[∫ t∧Tn

0

∫Rn

2m2−1|ξi|mKω,s(dξ)ds

].

The inequality∫ t∧Tn

0Y

m2s 1s<T∆ds ≤ n

m2 ∧ Y

m2t∧Tn follows from the fact that Ys

is non-decreasing and Ys− ≤ n for s ≤ Tn. The last jump might cause thatYTn > n, but this is not felt in the integral. Choosing for example ε = 1

2C,

leads to

1

2Ex[Y

m2t∧Tn

]≤ Ex

[Y

m2t∧Tn

]− 1

2Ex[nm2 ∧ Y

m2t∧Tn

]≤ CEx

[∫ t∧Tn

0

(1 + ‖Xs1s<T∆‖m +

∫Rn|ξi|mKω,s(dξ)

)ds

],

which also remains true if one of both sides is infinite. Since Ym2t∧Tn ≤ Y

m2t∧Tn+1

,


we can apply the monotone convergence theorem to obtain

Ex[Y

m2t

]≤ C

(1 +

∫ t

0

Ex[‖Xs1s<T∆‖m

]ds

+

∫ t

0

Ex[∫

Rn|ξi|mKω,s(dξ)

]ds

).

Again by (4.17) and the equivalence of the 1- and 2- norms, we finally get thedesired estimate (4.16).

4.3 Characterization by means of the Extended

Generator

We are finally prepared to prove the implication (iv)⇒ (iii) of Theorem 4.1.8for m ≥ 2 an even number. Based on the maximal inequality stated inLemma 4.2.3 above, we shall exploit the fact that under the assumption ofTheorem 4.1.8 (iv) we have Ex

[sups≤t

∥∥Xs1s<T∆∥∥m] <∞ whenever

Ex[∥∥Xs1s<T∆

∥∥m] <∞, for all s ≤ t.

4.3.1 Proof of Theorem 4.1.8 (iv) ⇒ (iii)

Proof. Recall that M was defined by

M fs = f(Xs)− f(X0)−

∫ s

0

Gf(Xu)du.

We now show that

Ex[sups≤t

∣∣M fs

∣∣] = Ex[sups≤t

∣∣∣∣f(Xs)− f(X0)−∫ s

0

Gf(Xu)du

∣∣∣∣] <∞for every fixed t ≥ 0. The dominated convergence theorem then implies thatM f

t is a (Ft,Px)-martingale and thus Af = Gf on Pol≤m(S). Since we haveGf ∈ Pol≤m(S) for f ∈ Pol≤m(S), we can dominate sups≤t

∣∣M fs

∣∣ with the

4.3. Characterization by means of the Extended Generator 133

following expression:

sups≤t

∣∣M fs

∣∣ ≤ sups≤t

∣∣f(Xs)1s<T∆∣∣+ |f(X0)|+ sup

s≤t

∣∣∣∣∫ s

0

Gf(Xu)1u<T∆du

∣∣∣∣≤ C1

(1 + sup

s≤t

∥∥Xs1s<T∆∥∥m)+ C2 (1 + ‖X0‖m)

+

∫ t

0

C3

(1 +

∥∥Xs1s<T∆∥∥m) ds

≤ C

(1 + ‖X0‖m + sup

s≤t

∥∥Xs1s<T∆∥∥m) ,

with C = C1 + C2 + C3t for some constants C1, C2, C3. Due to Proposi-tion 4.2.1, (Xt1t<T∆

) satisfies the condition of Lemma 4.2.3 above. Moreover,by the moment assumption on X, that is, t 7→ Ptf(x) is well-defined for allf ∈ Pol≤m(S), and the fact that m is even, Lemma 4.2.3 yields

Ex[sups≤t

∥∥Xs1s<T∆∥∥m] <∞

for every t ≥ 0. Indeed, since m is even, this follows from (4.10), where wehave

Ex[∫

Rn‖ξ‖mKω,t(dξ)

]≤ C

(1 + Ex

[‖Xt1t<T∆‖m

]).

Finally, by the above estimate, we have Ex[sups≤t

∣∣M fs

∣∣] <∞ for every t ≥ 0,proving the assertion.

Remark 4.3.1. If m > 2 is an odd number, the above proof implies that theassumptions of Theorem 4.1.8 (iv) together with

Ex[∫


]<∞

are sufficient for X being an m-polynomial process. In particular, the condi-tions of Theorem 4.1.8 (iv) are always sufficient in the case of pure diffusionsand jumps supported on the positive orthant, where the latter statement is aconsequence of (4.9) and the moment conditions on X.

4.3.2 Semimartingales which are Polynomial Processes

Corollary 4.3.2. Let X be a time-homogeneous Markov process with statespace S∆ and semigroup (Pt) such that t 7→ Ptf(x) is continuous at t = 0


for all f ∈ Pol≤m(S) with m ≥ 2. Furthermore, suppose that (Xt1t<T∆) is asemimartingale, whose characteristics (B,C, ν) with respect to the “truncationfunction” χ(ξ) = ξ satisfy the condition of Proposition 4.2.1. If

Px[t < T∆] = e−γt (4.22)

for some constant γ ≥ 0 and if Ex[∫Rn ‖ξ‖

mKω,t(dξ)] < ∞, then X is anm-polynomial process.

Proof. By Condition (4.22) and Theorem 4.1.8, X is 0-polynomial, since

Ex[1t<T∆ − 1 +

∫ t

0

γ1s<T∆ds

]= Px [t < T∆]− 1 +

∫ t

0

γPx [s < T∆] ds

= e−γt − 1 +

∫ t

0

γe−γsds = 0.

Moreover, as a consequence of (4.7) and (4.9),∫Rn ‖ξ‖

kKω,t(dξ) < ∞ for all2 ≤ k ≤ m and t ≥ 0. Hence, for all 2 ≤ l ≤ k ≤ m,(∫ t

0

‖Xs‖k−l1s<T∆

(∫Rn‖ξ‖lKω,s(dξ)

)ds

)t

(4.23)

is of locally integrable variation, as it is a predictable finite variation process.This implies in particular that (Xt1t<T∆) is a special semimartingale andjustifies the choice of the “truncation function” χ(ξ) = ξ. An application ofIto’s formula to f(x) = xk for 1 ≤ k = |k| ≤ m, and the fact that (4.23) is oflocally integrable variation then implies that

f(Xt)1t<T∆ − f(X0)−n∑i=1

∫ t

0

Dif(Xs)1s<T∆bi(Xs)ds

+1

2

n∑i,j=1

∫ t

0

(Dijf(Xs)1s<T∆

(cs,ij +

∫RnξiξjKω,s(dξ)

))ds

+

∫ t

0

∫Rn

k∑|l|=3

(k

l

)Xk−ls 1s<T∆ξ

l

Kω,s(dξ)ds

(4.24)

is a local martingale. The extended infinitesimal generator G applied to xk isthus given by the last three terms in equation (4.24), which by our assump-tions clearly map Pol≤k(S) into Pol≤k(S) for 1 ≤ k ≤ m. Hence we are inthe situation of Theorem 4.1.8 (iv). The assertion then follows by using thesame arguments as in the proof of Theorem 4.1.8 (iv) ⇒ (iii) above and theassumption that Ex[

∫Rn ‖ξ‖

mKω,t(dξ)] <∞ for all t ≥ 0.

4.3. Characterization by means of the Extended Generator 135

Remark 4.3.3. (i) Let us mention that the characteristics of (Xt1t<T∆)in Corollary 4.3.2 and Proposition 4.2.1 are specified with respect to the“truncation function” χ(ξ) = ξ. While C and ν do not depend on thischoice, the characteristic B does depend on χ. If one chooses anothertruncation function χ′, then the difference between B and B′ is givenby∫ t

0

∫Rn\0 (χ′(ξ)− χ(ξ)) ν(ds, dξ). Thus the requirement that C and

ν are as in Corollary 4.3.2 and(bt1t<T∆ +

∫Rn\0

(χ(ξ)− χ′(ξ))Kω,t(dξ)

)=

1∑|k|=0

αkXkt 1t<T∆

(4.25)

is an equivalent condition guaranteeing that X is m-polynomial.

(ii) We now give two examples of random measures Kω,t(dξ) which satisfythe conditions of Proposition 4.2.1 as long as cij ∈ Pol≤2(S) and b sat-isfies (4.25).

(a) The first one essentially requires Kω,t(dξ) = K(Xt(ω), dξ) to be aquadratic polynomial in Xt(ω), that is,

Kω,t(dξ) =

(µ00(dξ)

‖ξ‖2 ∧ 1+

n∑i=1

Xt,iµi0(dξ)

‖ξ‖2 ∧ 1+∑i≤j

Xt,iXt,jµij(dξ)

‖ξ‖2 ∧ 1

),

where all µij are finite signed measures on Rn such that K(x, ·) isa well-defined Levy measure for every x ∈ S. Denoting the Jordandecomposition of µij by µ+

ij, µ−ij, it is necessary to require∫

‖ξ‖>1

‖ξ‖m(µ+ij(dξ) + µ−ij(dξ)) =

∫‖ξ‖>1

‖ξ‖m|µij(dξ)| <∞.

(b) Alternatively, K can be specified as the pushforward of a Levy mea-sure under an affine function. Let d ≥ 1 and let

g : S × Rd → Rn, (x, y) 7→ g(x, y) = gx(y) = H(y)x+ h(y),

be an affine function in x. Here, H : Rd → Rn×n and h : Rd → Rn

are assumed to be measurable. We then define K by

Kω,t(dξ) = K(Xt(ω), dξ) := (gXt)∗µ(dξ),

where for each x ∈ S, (gx)∗µ denotes the pushforward of the mea-sure µ under the map gx. Moreover, µ is a Levy measure on Rd

integrating∫Rd\0

(‖H(y)‖k + ‖h(y)‖k

)µ(dy) for all 1 ≤ k ≤ m.


4.4 Examples

Example 4.4.1 (Affine processes). Every affine process X on S = Rp+×Rn−p

is m-polynomial if the killing rate γ is constant, if the Levy measures µi,i = 1, . . . , p, satisfy

∫‖ξ‖>1

‖ξ‖µi(dξ) < ∞, and if Ptf(x) is well-defined for

f ∈ Pol≤m(S).

Proof. The characteristic function of an affine process is given by

Ex[e〈iu,Xt〉

]= eφ(t,iu)+〈ψ(t,iu),x〉,

where φ and ψ are solutions of ordinary differential equations. The assump-tions on the killing rate and the jump measure imply that ψ(t, 0) = 0 forall t ≥ 0 (see Duffie et al. [2003, Proposition 9.1, Lemma 9.2]), whence Xis 0-polynomial. By our assumptions, moments up to order m exist, whichimplies that the characteristic function is m-times continuously differentiable.A computation shows that for all k ≤ m, the kth derivative is a polynomial inx of degree k. Continuity of t 7→ Ptf(x) follows from the fact that the deriva-tives of φ and ψ with respect to u are also solutions of ordinary differentialequations.

Note that the explicit knowledge of φ and ψ is not necessary to computethe moments of an affine process. Simply the knowledge of its characteristics,which determine the linear map A, is enough.

Example 4.4.2 (Levy processes). Let L be a Levy process on Rn with triplet(b, c, µ) satisfying

∫‖ξ‖>1

‖ξ‖mµ(dξ) <∞. Then the Markov process X = x+L

is m-polynomial.

Example 4.4.3 (Exponential Levy models). Exponential Levy models areof the form X = xeL, where L is a Levy process on R with triplet (b, c, µ).Under the integrability assumption

∫|y|>1

emyµ(dy) < ∞, which guarantees

the existence of Ex [|Xt|m], exponential Levy models are m-polynomial, sincewe have Ex

[xmemLt

]= xmetψ(m), where ψ denotes the cumulant generating

function of the Levy process.

In order to apply Theorem 4.1.8 (iv) ⇒ (i) together with Remark 4.3.1 orCorollary 4.3.2 to the following examples, we assume m ≥ 2, i.e., the processX admits moments up to order 2 at least.

Example 4.4.4 (Levy driven SDEs). Let L denote a Levy process on Rd withtriplet (b, c, µ). Suppose furthermore that V1, . . . , Vd are affine functions, i.e.,

4.4. Examples 137

we have Vi : S → Rn, x 7→ Hix+hi, where Hi ∈ Rn×n and hi ∈ Rn. A processX which solves the stochastic differential equation

dXt =d∑i=1

Vi(Xt−)dLt,i, X0 = x ∈ S,

and leaves S invariant, is m-polynomial, if t 7→ Ptf(x) is continuous for x ∈ Sand f ∈ Pol≤m(S). This is for instance the case if moment conditions on theLevy measure of the type ∫

‖y‖>1

‖y‖m+εµ(dy) <∞ (4.26)

hold true for some ε > 0.

Proof. For C2-functions and general Lipschitz continuous functions V1, . . . , Vdthe extended infinitesimal generator of X with respect to some truncationfunction χ is given by

1

2

n∑i,j=1

((V1(x) . . . Vd(x))c(V1(x) . . . Vd(x))′)ijDijf(x)

+

⟨d∑i=1

Vi(x)bi,∇f(x)

⟩

+

∫ (f

(x+

d∑i=1

Vi(x)yi

)− f(x)−

⟨d∑i=1

Vi(x)χi(y),∇f(x)

⟩)µ(dy).

Concerning the random measure Kω,t(dξ), this example corresponds to the

situation of Remark 4.3.3 (ii) (b) with g(x, y) = H(y)x+h(y) =∑d

i=1Hiyix+hiyi. The condition of Remark 4.3.1, that is,

Ex[∫


]= Ex

[∫Rn

∥∥∥∥∥d∑i=1

HiyiXt + hiyi

∥∥∥∥∥m

µ(dy)

]<∞,

is satisfied due to (4.26). Moreover, the existence of the (m+ ε)th-moment ofthe measure µ, that is, Condition (4.26), is sufficient for Ex[‖Xt‖m+ε] to befinite (see Protter [2005, Theorem V.67] or Jacod et al. [2005]), which then alsoimplies continuity of t 7→ Ptf(x) for all f ∈ Pol≤m(S). Hence Theorem 4.1.8(iv) ⇒ (i) and Remark 4.3.1 yield the assertion.


Example 4.4.5 (Quadratic term structure models Chen, Filipovic, and Poor[2004]). Consider the following quadratic short rate model r, specified as non-negative quadratic function of a one-dimensional Ornstein-Uhlenbeck processY

rt = R0 +R1Yt +R2Y2t ,

for appropriate Ri ∈ R. Here, Y is given by

dYt = (b+ βYt)dt+ σdWt, b, β ∈ R, σ ∈ R+,

where W is a standard Brownian motion. The joint process X = (Y, r) thenfollows the dynamics(dYtdrt

)=

((b

R1b+R2σ2 − 2R0β

)+

(β

2R2b−R1β

)Yt +

(0

2β

)rt

)dt

+

(σ

(R1 + 2R2Yt)σ

)dWt.

Due to the implication (iv) ⇒ (i) of Theorem 4.1.8, it is thus a polynomialprocess with

Ct =

∫ t

0

(σ2

(1 R1

R1 R21

)+ σ2

(0 2R2

2R2 4R1R2

)Ys + σ2

(0 00 4R2

2

)Y 2s

)ds.

Example 4.4.6 (Jacobi process). Another example of a polynomial process isthe Jacobi process (see Gourieroux and Jasiak [2006]), which is the solutionof the stochastic differential equation

dXt = −β(Xt − θ)dt+ σ√Xt(1−Xt)dWt, X0 = x ∈ [0, 1],

on S = [0, 1], where θ ∈ [0, 1] and β, σ > 0. This example can be extended byadding jumps, where the jump times correspond to those of a Poisson processN with intensity λ and the jump size is a function of the process level. Indeed,if a jump occurs, then the process is reflected at 1

2so that it remains in the

interval [0, 1], i.e., the (extended) infinitesimal generator is given by

Gf =1

2σ2(x(1− x))

d2f(x)

dx2− β(x− θ)df(x)

dx+ λ(f(1− x)− f(x)).

By applying Theorem 4.1.8 (iv)⇒ (i), it is thus easily seen that this process ispolynomial. In terms of Remark 4.3.3 (ii) (b), we have here g(x, y) = −2yx+yand µ(dy) = λδ1(dy).

4.4. Examples 139

Example 4.4.7 (Pearson diffusions). Example 4.4.6 (without jumps) as wellas the Ornstein-Uhlenbeck and the Cox-Ingersoll-Ross process, all of them withmean-reverting drift, can be subsumed under the class of the so-called Pearsondiffusions, which are solutions of SDEs of the form

dXt = −β(Xt − θ)dt+√

(a+ α10Xt + α11X2t )dWt, X0 = x, (4.27)

where β > 0 and α10, α11 and a are specified such that the square root iswell-defined. Forman and Sørensen [2008] give a complete classification ofthe different types of the Pearson diffusion in terms of their invariant distri-butions. Note that existence of (weak) solutions of (4.27) follows from con-tinuity and linear growth of the drift and the diffusion part (see Rogers andWilliams [1987, Theorem V.23.5]).

Example 4.4.8 (Dunkl process). The (extended) infinitesimal generator ofthe so-called Dunkl process (see Dunkl [1992], Gallardo and Yor [2006]) isgiven by

Gf(x) =d2f(x)

dx2+

λ

2x2

∫R

(f(x+ ξ)− f(x)− ξ df(x)

dx

)δ−2x(dξ)

=d2f(x)

dx2+λ

x

df(x)

dx+λ(f(−x)− f(x))

2x2.

Since Kω,t(dξ) = λ2X2

tδ−2Xt(dξ) and since

Ex[∫

R|ξ|mKω,t(dξ)

]= Ex

[λ| − 2Xt|m

2X2t

]<∞

for m ≥ 2, we derive from Theorem 4.1.8 and Remark 4.3.1 that the Dunklprocess is a polynomial process.

Part III

Applications

141

Chapter 5

Multivariate Affine StochasticVolatility Models

In this chapter we study applications of affine processes on the symmetric coneof positive semidefinite d × d matrices, denoted by S+

d and always assumed > 1. These matrix-valued affine processes have arisen from a large andgrowing range of useful applications in finance, including multi-asset optionpricing with stochastic volatility and correlation structures, and fixed-incomemodels with stochastically correlated risk factors and default intensities. Here,we focus on multivariate affine stochastic volatility models.

For illustration, let us consider the following model consisting of a d-dimensional logarithmic price process with risk-neutral dynamics

dYt =

(r1− 1

2Xdiagt

)dt+

√XtdVt, Y0 = y, (5.1)

and stochastic covariation process X = 〈Y, Y 〉. Here, V denotes a standard d-dimensional Brownian motion, r the constant interest rate, 1 the vector whoseentries are all equal to one and Xdiag the vector containing the diagonal entriesof X.

The necessity to specify X as a process in S+d such that it qualifies as

covariation process is one of the mathematically interesting aspects of suchmodels. Beyond that, the modeling of X must allow for enough flexibility inorder to reflect the stylized facts of financial data and to adequately capturethe dependence structure of the different assets. If these requirements are met,the model can be used as a basis for financial decision-making in the area ofmulti-asset options pricing and hedging of correlation risk.

The tractability of such a model, for example for computing prices ofEuropean derivatives, crucially depends on the dynamics of X. A large part

143

144 Chapter 5. Multivariate Affine Volatility Models

of the literature in the area of multivariate stochastic volatility modeling hasproposed the following affine dynamics for X

dXt = (b+MXt +XtM>)dt+

√XtdWtΣ + Σ>dW>

t

√Xt + dJt,

X0 = x ∈ S+d ,

(5.2)

where b is some suitably chosen matrix in S+d , M, Σ some invertible matrices,

W a standard d× d matrix of Brownian motions possibly correlated with V ,and J a pure jump process whose compensator is an affine function of X. Thisaffine multivariate stochastic volatility model generalizes the well-known one-dimensional models of Heston [1993], for the diffusion case, and of Barndorff-Nielsen and Shephard [2001], for the pure jump case. It is tractable in thesense that the characteristic function of (X, Y ) is known up to the solution ofan ordinary differential equation.

In the absence of jumps, the process described by (5.2) is a so-calledWishart process, as its marginal distributions are the non-central Wishartdistributions described in Section 3.5 and Section 3.7. These processes werefirst introduced by Bru [1989, 1991] and have subsequently been applied inmathematical finance.

Gourieroux and Sufana [2003, 2004] and Gourieroux, Monfort, and Sufana[2005] seem to be the first who considered these processes for term structuremodeling. Other financial applications thereof have then been taken up andcarried further by various authors, including Da Fonseca, Grasselli, and Ielpo[2007a, 2009], Da Fonseca, Grasselli, and Tebaldi [2007b, 2008], Buraschi, Cies-lak, and Trojani [2007] and Buraschi, Porchia, and Trojani [2010]. Barndorff-Nielsen and Stelzer [2007, 2011] provided a theory for a certain class of matrix-valued Levy driven Ornstein-Uhlenbeck processes of finite variation. Theseprocesses have then been considered for multivariate stochastic volatility mod-eling in Pigorsch and Stelzer [2009a] and Muhle-Karbe, Pfaffel, and Stelzer[2010]. Leippold and Trojani [2008] introduced S+

d -valued affine jump diffu-sions and provided financial examples, including multivariate option pricing,fixed-income models and dynamic portfolio choice.

5.1 Definition

Analogously to the one dimensional case (see Keller-Ressel [2010, Section 2]),we define a multivariate affine stochastic volatility model via the joint mo-ment generating function of the logarithmic price process and the covariationprocess: We assume that the d-dimensional asset price process (St)t≥0 is givenby

St = ert+Yt , t ≥ 0,

5.1. Definition 145

where r is the constant nonnegative interest rate and (Yt)t≥0 the d-dimensionaldiscounted logarithmic price process starting at Y0 = y ∈ Rd a.s. The dis-counted price process is thus simply (eYt)t≥0. Therefore we shall assume inthe sequel that r = 0, and that (St)t≥0 is already discounted. Let (Xt)t≥0

denote the stochastic covariation process, which takes values in S+d , the cone

of positive semidefinite d × d matrices and starts at X0 = x ∈ S+d a.s. In

order to qualify for a multivariate affine stochastic volatility model, the jointprocess (Xt, Yt)t≥0 with state space D := S+

d ×Rd has to satisfy the followingassumptions:

A1) (Xt, Yt)t≥0 is a stochastically continuous time-homogeneous Markov pro-cess on D = S+

d × Rd.

A2) The Fourier-Laplace transform of (Xt, Yt) has exponential affine depen-dence on the initial states (x, y), that is, there exist functions (t, u, v) 7→Φ(t, u, v) and (t, u, v) 7→ Ψ(t, u, v) such that

Ex,y[etr(uXt)+v>Yt

]= Φ(t, u, v)etr(Ψ(t,u,v)x)+v>y (5.3)

for all (x, y) ∈ D and all (t, u, v) ∈ Q where

Q =

(t, u, v) ∈ R+ × Sd + iSd × Cd∣∣∣

Ex,y[∣∣∣etr(uXt)+v>Yt

∣∣∣] = Ex,y[etr(Re(u)Xt)+Re(v)>Yt

]<∞

.

Remark 5.1.1. (i) Since we shall establish conditions under which S is amartingale, we need in particular that Ex,y[eYt,i ] <∞ for i ∈ 1, . . . , d.For this reason we here suppose – in contrast to Definition 1.1.4 – thatthe affine property holds on the set Q ⊇ R+×U and not only on R+×U .Note that in this case the set U corresponds to S−d + iSd × iRd.

(ii) Due to the term v>y in the moment generating function, we here do notconsider all affine processes on S+

d ×Rd. For stochastic volatility modelsit is however reasonable to suppose this form of the moment generatingfunction. It simply means that Yt with Y0 = y is shifted by c for everyt ≥ 0 if we start at y + c. Note that Assumption A2) also implies thatthe covariation process X is a Markov process on S+

d in its own rightsuch that we can apply the theory developed in Chapter 3.

(iii) For the moment, we do not make the assumption that (St) is conservativeor a martingale. Instead this will be established in the following section.


(iv) A possible extension of affine stochastic volatility models, as introducedabove, is to assume that the interest rate r is described by an affine shortrate model (see Duffie et al. [2003, Chapter 11]).

The following theorem summarizes some important properties of multivari-ate affine stochastic volatility models and is based on the results of Chapter 1.For the sake of readability, we state the proof nevertheless by using the samearguments as in Proposition 1.1.6 (iv) and applying Theorem 1.5.4.

Theorem 5.1.2. Let (τ, u, v) ∈ Q for some τ ≥ 0 and suppose that

E0,0

[etr(uXτ )+v>Yτ

]6= 0. (5.4)

Then, for t, s ≥ 0 such that t+ s = τ , we have (t, u, v) ∈ Q, (s,Ψ(t, u, v), v) ∈Q and

E0,0

[etr(uXt)+v>Yt

]6= 0, and E0,0

[etr(Ψ(t,u,v)Xs)+v>Ys

]6= 0. (5.5)

Moreover, the functions Φ and Ψ satisfy the semiflow equations, that is,

Φ(t+ s, u, v) = Φ(t, u, v)Φ(s,Ψ(t, u, v), v),

Ψ(t+ s, u, v) = Ψ(s,Ψ(t, u, v), v),(5.6)

and the derivatives

F(u, v) :=∂Φ(t, u, v)

∂t

∣∣∣∣∣t=0

and R(u, v) :=∂Ψ(t, u, v)

∂t

∣∣∣∣∣t=0

(5.7)

exist and are continuous in (u, v). Furthermore, for t ∈ [0, τ), Φ and Ψ satisfythe generalized Riccati equations

∂tΦ(t, u, v) = Φ(t, u, v)F(Ψ(t, u, v), v), Φ(0, u, v) = 1, (5.8)

∂tΨ(t, u, v) = R(Ψ(t, u, v), v), Ψ(0, u, v) = u. (5.9)

Proof. By the Markov property and the law of iterated expectations, we havefor all t, s ≥ 0 such that t+ s = τ

Φ(t+ s, u, v)etr(Ψ(t+s,u,v)x)+v>y = Ex,y[etr(uXt+s)+v>Yt+s

]= Ex,y

[Ex,y

[etr(uXt+s)+v>Yt+s

∣∣∣Fs]]= Ex,y

[EXs,Ys

[etr(uXt)+v>Yt

]]. (5.10)

5.2. Form of F and R 147

Hence (t, u, v) ∈ Q and (s,Ψ(t, u, v), v) ∈ Q. Moreover, by Assumption (5.4),neither the inner nor the outer expectation can be 0, which thus implies (5.5).We can therefore write (5.10) as

Ex,y[etr(uXt+s)+v>Yt+s

]= Ex,y

[Φ(t, u, v)etr(Ψ(t,u,v)Xs)+v>Ys

]= Φ(t, u, v)Φ(s,Ψ(t, u, v), v)etr(Ψ(s,Ψ(t,u,v),v)x)+v>y

for all (x, y) ∈ S+d × Rd. Combining this with the left hand side of (5.10),

already implies (5.6).

Differentiability of Φ(t, u, v) and Ψ(t, u, v) at t = 0 and continuity of F(u, v)and R(u, v) in (u, v) follow from Theorem 1.5.4. Note that the proof works ex-actly alike for (u, v) /∈ U . Due to this property, we are allowed to differentiatethe equations (5.6) with respect to t and evaluate them at 0. As a consequence,Φ and Ψ satisfy the generalized Riccati equations (5.8) and (5.9).

5.2 Form of F and R

Similar to Section 2.3.1 and 2.3.2, we are interested in the particular formof the functions F and R and the restrictions on the involved parameters.From Theorem 1.5.4 it already follows that F and R have parametrizationof Levy Khintchine type, but in order to obtain the particular admissibilityconstraints in this setting, we proceed similarly as in Proposition 2.3.2 andProposition 2.3.3. Note that we cannot directly apply the results of Chapter 2and Chapter 3, since we are here working on the mixed state space S+

d × Rd.

Before stating the theorem, let us first introduce some notation. We definethe following set

V = (u, v) ∈ Sd + iSd × Cd | ∃ t > 0 such that (t, u, v) ∈ Q.

Note that U ⊆ V . Moreover, for an element in z ∈ Sd × Rd, we write zX forthe components in Sd and zY for those in Rd. In particular, χY : Rd → Rd

denotes some bounded continuous truncation functions.

Theorem 5.2.1. For (u, v) ∈ V, the functions F and R are of Levy-Khintchine


form, that is,

F(u, v) =1

2v>aY v + tr(bXu) + v>bY − c

+

∫D

(etr(ξXu)+v>ξY − 1− v>χY (ξY )

)m(dξ), (5.11)

R(u, v) = 2uαXu+1

2QY (v) + L(u, v) +B>X(u) +B>Y (v)− γ

+

∫D

(etr(ξXu)+v>ξY − 1− v>χY (ξY )

)µ(dξ), (5.12)

where

αX ∈ S+d , (5.13)

QY : Rd → S+d is a quadratic function, (5.14)

L : Sd × Rd → Sd is a bilinear function such that (5.15)

4uαXu+QY (v) + 2L(u, v) ∈ S+d , ∀(u, v) ∈ Sd × Rd, (5.16)

µ is an S+d -valued Borel measure with supp(µ) ⊆ D such that, for every

x ∈ S+d , M(x, dξ) = tr(µ(dξ)x) is a Levy measure, which satisfies∫

D

((√tr(ξ2

X) + ξ>Y ξY

)∧ 1

)M(x, dξ) <∞, (5.17)

and for all x ∈ S+d and (u, v) ∈ V∫

D∩√

tr(ξ2X)+ξ>Y ξY >1

etr(Re(u)ξX)+Re(v)>ξYM(x, dξ) <∞, (5.18)

B>X : Sd → Sd, B>Y : Rd → Sd are linear maps such that, for all u ∈ S−d ,

x ∈ S+d with tr(ux) = 0,

tr(B>X(u)x) ≤ 0, (5.19)

and

γ ∈ S+d , (5.20)

aY ∈ S+d , (5.21)

bX − (d− 1)αX ∈ S+d , bY ∈ Rd, (5.22)

c ∈ R+. (5.23)


Finally, m is a Borel measure with supp(m) ⊆ D such that∫D

((√tr(ξ2

X) + ξ>Y ξY

)∧ 1

)m(dξ) <∞, (5.24)∫

D∩√

tr(ξ2X)+ξ>Y ξY >1

etr(Re(u)ξX)+Re(v)>ξYm(dξ) <∞, ∀(u, v) ∈ V . (5.25)

Proof. We proceed similarly as in the proof of Proposition 2.3.2. Note that

the t-derivative of Ex,y[etr(uXt)+v>Yt

]at t = 0 exists for all (x, y) ∈ D and

(u, v) ∈ V , since

limt↓0

Ex,y[etr(uXt)+v>Yt

]− etr(ux)+v>y

t

= limt↓0

Φ(t, u, v)etr(Ψ(t,u,v),x)+v>y − etr(ux)+v>y

t

= (F(u, v) + tr(R(u, v)x)etr(ux)+v>y

is well-defined by Theorem 5.1.2. Moreover, we can also write

(F(u, v) + tr(R(u, v)x)

= limt↓0

Ex,y[etr(uXt)+v>Yt

]− etr(ux)+v>y

tetr(ux)+v>y

= limt↓0

1

t

(∫D

etr(u(ξX−x))+v>(ξY −y)pt(x, y, dξ)− 1

)= lim

t↓0

(1

t

∫D−(x,y)

(etr(uξX)+v>ξY − 1

)pt(x, y, dξ + (x, y))

+pt(x, y,D)− 1

t

),

where pt(·, ·, dξ) denotes the Markov kernels. By the above equalities and thefact that pt(x, y,D) ≤ 1, we then obtain for (u, v) = 0

0 ≥ limt↓0

pt(x, y,D)− 1

t= F(0, 0) + tr(R(0, 0), x).

Setting −F(0, 0) = c and −R(0, 0) = γ yields (5.23) and (5.20). We thus have

(F(u, v) + c) + tr((R(u, v) + γ)x)

= limt↓0

1

t

∫D−(x,y)

(etr(uξX)+v>ξY − 1

)pt(x, y, dξ + (x, y)) (5.26)


By the same arguments as in the proof of Proposition 2.3.2, it follows thatthe left hand side of (5.26) is the logarithm of the Fourier-Laplace transformof some infinitely divisible probability distribution K(x, y, dξ) supported on(S+

d − R+x× Rd).In particular, for x = 0, K(0, y, dξ) is an infinitely divisible distribution

with support on D which cannot depend on y. Defining a scalar product onSd × Rd by 〈(u, v), (x, y)〉 := tr(ux) + v>y, we have by the Levy–Khintchineformula

F(u, v) + c =1

2〈(u, v), a(u, v)〉+ 〈(bX , bY ), (u, v)〉

+

∫Sd×Rd

(e〈(ξX ,ξY ),(u,v)〉 − 1− 〈(χX(ξ), χY (ξ)), (u, v)〉

)m(dξ),

where a is a positive semidefinite linear operator on S+d ×Rd, bX ∈ Sd, bY ∈ Rd,

m a Levy measure with support on Sd×Rd, χX an Sd-valued and χY and Rd-valued truncation function. Note that if (u, v) ∈ V , then (Re(u),Re(v)) ∈ Vas well and in particular F(Re(u),Re(v)) is well defined. Therefore

E[e〈(Re(u),Re(v),(LX1 ,L

Y1 )〉]<∞.

Here, (LX , LY ) corresponds to the Levy process associated to K(0, y, dξ). Thisin turn implies ∫

Sd×Rd∩‖ξ‖>1e〈(ξX ,ξY ),(Re(u),Re(v))〉m(dξ) <∞. (5.27)

Let now T denote the projection of S+d × Rd on S+

d . We can then applythe same arguments as in the proof of Proposition 2.3.3. Indeed, by the Levy-Khintchine formula on cones, the logarithm of the Fourier-Laplace transformof T∗K(0, y, dξ) is given by

F(u, 0) = tr(bXu) +

∫S+d

(etr(ξXu) − 1

)m(T−1(dξX)),

where bX ∈ S+d and m satisfies supp(T∗m) ⊆ S+

d as well as∫S+d

(√tr(ξ2

X) ∧ 1

)m(T−1(dξX)) <∞,

which together with (5.27) yields (5.24) and (5.25). In particular, we canchoose χX to be 0 and χY only to depend on ξY . Moreover, we have TaT> = 0


and since a is a positive semidefinite operator on S+d × Rd, we obtain Condi-

tion (5.21).Using again the same arguments as in the proof of Proposition 2.3.2, we

conclude that exp(tr((R(u, v)+γ)x)) is the Fourier-Laplace transform of someinfinitely divisible distribution L(x, y, dξ) on (S+

d − R+x)× Rd. Hence

tr((R(u, v) + γ)x)

=1

2〈(u, v), A(x)(u, v)〉+ 〈(BX(x), BY (x)), (u, v)〉

+

∫Sd×Rd

(e〈(ξX ,ξY ),(u,v)〉 − 1− 〈(χX(ξ), χY (ξ)), (u, v)〉

)M(x, dξ),

where, for each x ∈ S+d , A(x) is a positive semidefinite linear operator on

S+d × Rd, BX(x) ∈ Sd, BY (x) ∈ Rd, M(x, ·) is a Levy measure with support

on Sd × Rd, χX an Sd-valued and χY an Rd-valued truncation function. Bythe same arguments as before, we have∫

Sd×Rd∩‖ξ‖>1e〈(ξX ,ξY ),(Re(u),Re(v))〉M(x, dξ) <∞. (5.28)

We are again interested in the restrictions on the parameters. To this end,let now U denote the projection of Sd×Rd on Sd. We are then in the situationof the particular Euclidean Jordan algebra Sd, as discussed in Chapter 3 andSection 3.7. Using these results, we conclude that the logarithm of the Fourier-Laplace transform of U∗L(x, y, ·) is given by

tr((R(u, 0) + γ)x) =1

2tr(u(AX(x)u)) + tr(BX(x)u)

+

∫S+d

(etr(uξX) − 1

)M(x, U−1(dξX))

= 2 tr((uαXu)x) + tr(B>X(u)x)

+

∫S+d

(etr(uξX) − 1

)tr(µ(U−1(dξX))x).

(5.29)

Here, for each x ∈ S+d , AX(x) is a positive semidefinite operator on S+

d ,depending linearly on x and satisfying tr(u(AX(x)u)) = 4 tr((uαXu)x, whereαX ∈ S+

d , thus (5.13). Note here that the quadratic representation P (u)αXon the cone of positive semidefinite matrices is given by P (u)αX = uαXu.Moreover, due to Proposition 3.2.3 and since the rank d of Sd is supposed tobe greater than 1, we can choose χX(ξ) to be 0. Similarly as before χY (ξ)can also be chosen to depend only on ξY . By the linearity in x, we can write


M(x, dξ) = tr(µ(dξ)x) for the Levy measures M(x, ·), where µ is an S+d -valued

Borel measure with support on S+d × Rd. Concerning integrability, we have∫

S+d

(√tr(ξ2

X) ∧ 1

)M(x, U−1(dξX)) <∞,

which together with (5.28) implies (5.17) and (5.18). Furthermore, B>X : Sd →Sd is a linear map, which satisfies for all u ∈ S−d and x ∈ S+

d with tr(ux) = 0the following inward pointing drift condition

tr(B>X(u)x) ≤ 0,

hence implying (5.19). The remaining admissibility conditions follow from thefact that x 7→ 1

2〈(u, v), A(x)(u, v)〉 and x 7→ BY (x)v> are linear maps and A(x)

is positive semidefinite. Hence B>Y : Rd → Sd is linear and 12〈(u, v), A(x)(u, v)〉

can be written as

1

2〈(u, v), A(x)(u, v)〉 =

1

2tr(Q((u, v)), x),

where Q : Sd × Rd → S+d is a quadratic function. It can be decomposed into

Q((u, v)) = 4uαXu+QY (v) + 2L(u, v),

where QY : Rd → Sd is a quadratic function, that is, Condition (5.14), andL : Sd × Rd → Sd a bilinear function, that is, Condition (5.15), such thatQ((u, v)) ∈ S+

d for all (u, v) ∈ Sd×Rd, which implies (5.16). Finally, since foru ∈ S−d , F(u, 0) and R(u, 0) correspond to the derivatives of Φ and Ψ of anaffine process on S+

d , Proposition 3.2.6 implies the drift Condition (5.22).

Remark 5.2.2. Note the following consequences of Condition (5.16). Letx ∈ S+

d and u ∈ Sd such that ux = 0. Then

tr(L(u, v)x) = 0, (5.30)

for all v ∈ Rd. Indeed, in this case tr(4uαXux) = 0 and thus tr(QY (v)x) +2 tr(L(u, v)x) ≥ 0 for all v ∈ Rd. Hence

− tr(QY (v)x) ≤ 2 tr(L(u, v)x) ≤ tr(QY (v)x).

Dividing by ‖v‖ and letting v → 0, then yields (5.30). Note here that we haveux = 0 if u ∈ S−d , x ∈ S+

d and tr(ux) = 0 is satisfied.Moreover, if αX or QY is degenerate, we obtain by the same argument that

tr(L(u, v)x) = 0, (5.31)

5.3. Conservativeness and Martingale Property 153

for all (u, v) ∈ Sd×Rd for which we have tr(4uαXux) = 0 or tr(QY (v)x) = 0.In order to give some intuition for Condition (5.30), let us establish a

connection to the canonical state space Rm+ × Rn−m. Consider the linear dif-

fusion components, denoted by (α1, . . . , αm). Then the above condition onL(u, v) corresponds to the fact that αi,IJ has to satisfy a certain structurewhich guarantees that αi is positive semidefinite. We here use the notationof Duffie et al. [2003, Definition 2.6].

5.3 Conservativeness and Martingale Property

This section is devoted to investigate necessary and sufficient conditions suchthat Si = eYi , i = 1, . . . , d, is a martingale. Under such conditions S mayserve as price process under the risk neutral measure in an arbitrage free assetpricing model. Before addressing this issue, we first clarify when the process(X, Y ) is conservative. To this end, we apply a result of Mayerhofer et al.[2011] to our setting. For the notion of quasi-monotonicity, which is neededhere, we refer to Definition 2.3.5

Lemma 5.3.1. The function u 7→ R(u, 0), defined in (5.12), is quasi-mono-tone increasing on S−d and locally Lipschitz continuous on S−−d . Moreover,for every u ∈ S−−d , there exists a unique global S−−d -valued solution Ψ(t, u, 0)of (5.9).

Proof. The first assertion follows from Proposition 2.3.7 and the second onefrom Proposition 2.4.3. Note here that, for u ∈ S−d , R(u, 0) = −R(−u) andΨ(t, u, 0) = −ψ(t,−u).

The following lemma is a reformulation of Mayerhofer et al. [2011, Propo-sition 3.3].

Lemma 5.3.2. Let T > 0 and u ∈ S−d . If g(t) : [0, T )→ S−d is a solution of

dg(t)

dt= R(g(t), 0), g(0) = u, (5.32)

then g(t) Ψ(t, u, 0) for all t < T , where denotes the partial order on Sdinduced by S+

d .

Proof. The proof relies on Mayerhofer et al. [2011, Corollary A.3]. Choosingf = R and Df = S−d , the assumptions of quasi-monotonicity, local Lips-chitz continuity and the condition Ψ(t, ·, 0) : S−−d → S−−d are satisfied dueto Lemma 5.3.1. By Mayerhofer et al. [2011, Corollary A.3], we thus haveg(t) Ψ(t, u, 0) for all u ∈ S−−d . Continuity of u 7→ Ψ(t, u, 0) then yields theassertion.


Proposition 5.3.3. Suppose (X, Y ) satisfies the Conditions A1) and A2).Then (X, Y ) is conservative if and only if F(0, 0) = 0 and g = 0 is the onlyS−d -valued solution of (5.32) with g(0) = 0. Moreover, each of these statementsimplies R(0, 0) = 0.

Proof. Using Lemma 5.3.1 and 5.3.2, we can adapt the proof of Mayerhoferet al. [2011, Theorem 3.4] to our setting without modification.

Remark 5.3.4. As already mentioned in Section 3.3.3, a sufficient conditionfor (X, Y ) to be conservative is c = 0, γ = 0 and∫

D∩‖ξ‖≥1‖ξ‖M(x, dξ)(dξ) <∞, ∀x ∈ S+

d ,

(see Duffie et al. [2003, Section 9]).

Henceforth, let us assume that (X, Y ) is a conservative process. Thenthe martingale property of Si = eYi , i ∈ 1, . . . , d, can be characterized asstated in Theorem 5.3.5 below. This is a generalization of Keller-Ressel [2010,Theorem 2.5.b] and Mayerhofer et al. [2011, Remark 4.5] to the setting ofmultivariate affine stochastic volatility models.

Theorem 5.3.5. Suppose (X, Y ) is conservative and satisfies the ConditionsA1) and A2). Let i ∈ 1, . . . , d. Then Si = eYi is a martingale if and only ifthe following conditions hold:∫

|e>i ξY |>1ee>i ξYm(dξ) <∞, (5.33)∫

|e>i ξY |>1ee>i ξYM(x, dξ) <∞, ∀x ∈ S+

d , (5.34)

F(0, ei) =1

2e>i aY ei + e>i bY +

∫ (ee>i ξY − 1− e>i χY (ξY )

)m(dξ) = 0, (5.35)

R(0, ei) =1

2QY (ei) +B>Y (ei) +

∫ (ee>i ξY − 1− e>i χY (ξY )

)µ(dξ) = 0,

(5.36)

and g = 0 is the only S−d -valued solution of

dg(t)

dt= R∗(g(t), 0), g(0) = 0, (5.37)

where, for u ∈ S−d , R∗(u, 0) is defined by

R∗(u, 0) = 2uαXu+ L(u, ei) +B>X(u) +

∫ (etr(ξXu) − 1

)ee>i ξY µ(dξ).

Here, ei, i = 1, . . . , d, denotes ith canonical basis vector of Rd.

5.3. Conservativeness and Martingale Property 155

Proof. Let us first suppose that Si = eYi is a martingale. Then Ex,y[ee>i Yt ] <∞

and (t, 0, ei) ∈ Q for all t ∈ R+. Thus, by Theorem 5.2.1, Conditions (5.25)and (5.18) are satisfied, which yield (5.33) and (5.34). Moreover, since

ee>i y = Ex,y

[ee>i Yt]

= Φ(t, 0, ei)etr(Ψ(t,0,ei)x)+e>i y,

we have Φ(t, 0, ei) = 1 and Ψ(t, 0, ei) = 0. Equations (5.8) and (5.9) thusimply F(0, ei) = 0 and R(0, ei) = 0. The latter then yields for u ∈ S−d

R(u, ei) = 2uαXu+1

2QY (ei) + L(u, ei) +B>X(u) +B>Y (ei)

+

∫ (etr(ξXu)+e>i ξY − 1− e>i χY (ξY )

)µ(dξ)

= 2uαXu+ L(u, ei) +B>X(u) +

∫ (etr(ξXu) − 1

)ee>i ξY µ(dξ)

+1

2QY (ei) +B>Y (ei) +

∫ (ee>i ξY − 1− e>i χY (ξY )

)µ(dξ)

= R∗(u, 0) + R(0, ei) = R∗(u, 0).

Moreover, R∗(u, 0) is of form (5.12) with corresponding parameters

α∗X = αX ,

B>∗X (u) = B>X(u) + L(u, ei),

µ∗(dξ) = ee>i ξY µ(dξ).

In particular, these parameters satisfy the conditions of Theorem 5.2.1, thatis, (5.13), (5.17) and (5.19). Concerning the latter, we have by (5.19) and (5.30)

tr(B>∗X (u)x) = tr(B>X(u)x) + tr(L(u, ei)x) ≤ 0,

for all u ∈ S−d with tr(ux) = 0. Thus the assertions of Lemma 5.3.1 hold truefor R∗(u, 0). Let now g be a (local) solution of (5.37) on some interval [0, T ),satisfying g(0) = 0 and taking values in S−d . By Lemma 5.3.2, we thereforehave Ψ∗(t, 0, 0) g(t) for all t < T , where

∂Ψ∗(t, u, 0)

∂t= R∗(Ψ∗(t, u, 0), 0), Ψ∗(0, u, 0) = u.

As R∗(u, 0) = R(u, ei), we conclude that Ψ∗(t, 0, 0) = Ψ(t, 0, ei) = 0, whichyields g = 0.

Concerning the other direction, we can argue as in Kallsen and Muhle-Karbe [2010, Corollary 3.4]. Indeed, eYi is a σ-martingale due to (5.33), (5.34),


(5.35) and (5.36). This is a consequence of Kallsen [2003, Lemma 3.1] and thefact that eYi is a semimartingale, whose characteristics can be determined asstated in Kallsen [2006, Proposition 3]. Moreover, from Kallsen [2003, Propo-sition 3.1] it follows that it is a supermartingale, thus it is in particular inte-grable. Hence (t, 0, ei) ∈ Q for all t ∈ R+, and Φ(t, 0, ei) = 1 and Ψ(t, 0, ei) = 0would guarantee that eYi is a martingale. Let now g(t) := Ψ∗(t, 0, 0) = 0be the only solution of (5.37). Since (5.36) implies R(u, ei) = R∗(u, 0) asshown above, we have Ψ∗(t, 0, 0) = Ψ(t, 0, ei) = 0 and Φ(t, 0, ei) = 1 followsfrom (5.35).

We are now prepared to give our full definition of a multivariate affinestochastic volatility model. For this purpose we add two further assumptionsto A1) and A2).

A3) The process (X, Y ) is conservative.

A4) The discounted price processes Si = eYi , i = 1, . . . , d, are martingales.

Definition 5.3.6. The process (X, Y ) is called a multivariate affine stochasticvolatility model if it satisfies the Assumptions A1 – A4.

5.4 Semimartingale Representation of Affine

Volatility Models

In order to address the issue of existence and uniqueness of affine stochasticvolatility models, the aim of this section is to establish a semimartingale rep-resentation of multivariate affine stochastic volatility models. In particular,we want to construct a Brownian motion such that (X, Y ) becomes a solutionof a generalized SDE with jumps. To this end, we first study the functionsQY and L, appearing in (5.16).

Lemma 5.4.1. (i) Let QY : Rd → S+d be a quadratic function. Then QY (v)

is given by

QY (v) =∑k,l

Gklvkvl, (5.38)

where Gkl = Glk ∈ Sd, k, l ∈ 1, . . . , d such that QY (v) ∈ S+d for all

v ∈ Rd. In other words, there exists an endomorphism G : S+d → S+

d

such thatQY (v) = G(v ⊗ v) = G(vv>).

5.4. Semimartingale Representation of Affine Volatility Models 157

(ii) Let L : Sd×Rd → Sd be a bilinear function such that (5.30) is satisfied.Then L is necessarily of the form

L(u, v) = u

(d∑i=1

Hivi

)+

(d∑i=1

H>i vi

)u, (5.39)

where Hi, i = 1, . . . d, are matrices in Rd×d.

Proof. It is clear that any quadratic function with values in S+d is of form (5.38).

Define now a symmetric bilinear function QY : Rd×Rd → Sd via polarization,that is,

QY (v, w) =1

2(QY (v + w)−QY (v)−QY (w)).

Then QY induces a linear map G : Rd ⊗ Rd → Sd such that QY (v, w) =G(v ⊗w). Identifying Rd ⊗Rd with the vector space of d× d matrices, yieldsthe assertion.

Concerning the second statement, consider a map L : Sd → Sd such that

tr(L(u)x) = 0 for all u, x ∈ S+d with tr(ux) = 0. (5.40)

Denote by eLt(u) the semigroup induced by ∂tg(t, u) = L(g(t, u)). Then (5.40)

implies that eLt and e−Lt are semigroups which map S+d into S+

d and thus

eLt(S+d ) = S+

d for all t ≥ 0. By Pigorsch and Stelzer [2009b, Theorem 3.3],

L is necessarily of the form L(u) = uH + H>u for some matrix H ∈ Rd×d.

Since, for fixed v, we have L(u) = L(u, v) for some L with the above property,the linearity of v 7→ L(u, v) then yields (5.39).

Lemma 5.4.2. Let L be of form (5.39) for some matrices Hi ∈ Rd×d, i =1, . . . d. Then

u2 + vv> + L(u, v) (5.41)

is positive semidefinite for all (u, v) ∈ Sd × Rd if and only if all entries of Hi

except the ith column are zero, that is,

Hi =

0 · · · ρ1 · · · 0...

. . ....

. . ....

0 · · · ρd · · · 0

, (5.42)

where ρ = (ρ1, . . . , ρd)> with |ρi| ≤ 1 and ρ>ρ ≤ 1. Thus (5.41) can be written

as

u2 + vv> + uρv> + vρ>u. (5.43)


Proof. Let us first show that positivity of (5.41) for all (u, v) ∈ Sd×Rd impliesthe particular form of the matrices Hi, i ∈ 1, . . . , d. We first set v = ei,where ei denotes the ith basis vector. By choosing u = ±eie>i , it is easilyseen that H ij

i = 0 for all j 6= i, and |H iii | ≤ 1. Taking u = ±εeke>k for

k 6= i and letting ε → 0, then yields Hkji = 0 for all j 6= i and |Hki

i | ≤ 1for all k ∈ 1, . . . , d. This already implies the above form of Hi for somevector ρi = (H1i

i , . . . , Hdii )>. In order to prove that ρ1 = ρ2 = · · · = ρd,

choose v = aei + bej and u = εId for some a, b ∈ R and ε > 0. By positivesemidefiniteness, we have

((ε2Id + vv> + ε((Hi +H>i )a+ (Hj +H>j )b)ij)2

= (ab+ εaHjii + εbH ij

j )2

≤ (ε2Id + vv> + ε((Hi +H>i )a+ (Hj +H>j )b)ii

× (ε2Id + vv> + ε((Hi +H>i )a+ (Hj +H>j )b)jj

= (ε2 + a2 + 2εaH iii )(ε2 + b2 + 2εbHjj

j ).

Dividing by ε and letting ε→ 0, this yields

a2b(Hjjj −H

jii ) + ab2(H ii

i −Hijj ) ≥ 0.

With an appropriate choice of a and b we therefore obtain H iii = Hji

j and

Hjjj = H ij

i . Since i, j was arbitrary, the above form of Hi, i = 1, . . . d, forsome vector ρ is proved and we can write

u2 + vv> + L(u, v) = u2 + vv> + uρv> + vρ>u.

It remains to show that ρ>ρ ≤ 1. Choosing v = −ρ and u = Id leads to

Id − ρρ> 0. (5.44)

Since the only non-zero eigenvalue of ρρ> is given by ρ>ρ, (5.44) is equivalentto ρ>ρ ≤ 1.

The other direction is obvious, since

u2 + vv> + uρv> + vρ>u = (uρ+ v)(uρ+ v)> + u(Id − ρρ>)u 0,

by (5.44).

Remark 5.4.3. Note that the assertion of Lemma 5.4.2 also holds true whenreplacing u ∈ Sd by u ∈ Rd×d. In this case, we extend the bilinear function Lto Rd×d × Rd in the obvious way by setting

L(u, v) = u

(d∑i=1

Hivi

)+

(d∑i=1

H>i vi

)u> (5.45)

and consider uu> + vv> + L(u, v) instead.


Corollary 5.4.4. Consider the quadratic function given by (5.16), that is,

Q((u, v)) = 4uαXu+QY (v) + 2L(u, v).

Assume that αX is given by αX = Σ>Σ for some Σ ∈ Rd×d and let QY :Rd → S+

d be a quadratic function such that QY (v) = G(vv>), where G is anendomorphism of S+

d of the form G : S+d → S+

d , a 7→ G(a) = gag> for somed× d matrix g. Then

Q((u, v)) = 4uΣ>Σu+ gvv>g> + 2L(u, v)

is positive semidefinite for all (u, v) ∈ Sd × Rd if and only if L(u, v) is of theform

L(u, v) = uΣ>ρv>g> + gvρ>Σu, (5.46)

for some ρ ∈ Rd with |ρi| ≤ 1 and ρ>ρ ≤ 1.

Proof. By Remark 5.2.2 and Lemma 5.4.1 (ii), positivity of Q((u, v)) impliesthat L is of form (5.39) for some matrices Hi, i = 1, . . . , d. We now setu = 2uΣ> and v = gv and suppose for the moment that Σ and g are invertible.Then

2L(u, v) = 2u

(d∑i=1

Hivi

)+ 2

(d∑i=1

H>i vi

)u

= u

(d∑i=1

Hivi

)+

(d∑i=1

H>i vi

)u> =: L(u, v),

where Hi =∑d

k=1 g−1ki Σ−>Hk. Therefore

Q((u, v)) = uu> + vv> + L(u, v).

By Remark 5.4.3 and Lemma 5.4.2, this expression is positive semidefinite forall (u, v) ∈ Rd×d×Rd if Hi, i = 1, . . . , d is of form (5.42) for some ρ ∈ Rd with

|ρi| ≤ 1 and ρ>ρ ≤ 1. We therefore have L(u, v) = uρv>+vρ>u>. Substitutingback thus yields (5.46). If Σ and g are degenerate, the necessary direction isa consequence of (5.31). The sufficient direction follows for general Σ and gas in the proof of Lemma 5.4.2, since

4uΣ>Σu+ gvv>g> + 2uΣ>ρv>g> + 2gvρ>Σu =

(2uΣ>ρ+ gv)(2uΣ>ρ+ gv)> + 4uΣ>(Id − ρρ>)Σu 0.


We are now prepared to prove the announced semimartingale representa-tion of multivariate stochastic volatility models.

Theorem 5.4.5. Let (X, Y ) be a multivariate affine stochastic volatility modelwith F and R of the form (5.8) and (5.9) such that αX = Σ>Σ for some d×dmatrix Σ. Moreover, suppose that QY : Rd → S+

d is given by QY (v) = gvv>g>

for some d× d matrix g. Then there exist a vector ρ ∈ Rd satisfying |ρi| ≤ 1and ρ>ρ ≤ 1 and – possibly on an enlargement of the probability space – a d×dmatrix of Brownian motions W , and Rd-valued Brownian motions V and Z,all mutually independent, such that (X, Y ) admits the following representation(

Xt

Yt

)=

(xy

)+

(BXt

BYt

)+

( ∫ t0

(√XsdWsΣ + Σ>dW>

s

√Xs

)∫ t0

(√aY dZs + g>

√XsdVs

) )

+

∫ t

0

∫D

(0

χY (ξY )

)(µ(X,Y )(ds, dξ)− ν(ds, dξ))

+

∫ t

0

∫D

(ξX

ξY − χY (ξY )

)µ(X,Y )(ds, dξ),

where

BXt =

∫ t

0

(bX +BX(Xs)) ds,

BYt,i = −

∫ t

0

(1

2aY,ii +

∫D

(eξY,i − 1− χY,i(ξY )

)m(dξ)

)ds

−∫ t

0

1

2(g>Xsg)iids

−∫ t

0

∫D

(eξY,i − 1− χY,i(ξY )

)M(Xs, dξ)ds,

Vt = (√

1− ρ>ρ)Vt +Wtρ,

ν(dt, dξ) = (m(dξ) +M(Xt, dξ))dt,

and µX,Y denotes the random measure associated with the jumps of (X, Y ).

Proof. Due to Definition 5.3.6, the functions F and R have to fulfill the con-ditions of Theorem 5.3.5. Furthermore, as αX = Σ>Σ and QY is given byQY (v) = gvv>g>, it follows from Corollary 5.4.4 that there exists some ρ ∈ Rd

satisfying |ρi| ≤ 1 and ρ>ρ ≤ 1 such that the positive semidefinite quadraticfunction Q((u, v)) defined in (5.16) is of form

Q((u, v)) = 4uΣ>Σu+ gvv>g> + 2uΣ>ρv>g> + 2gvρ>Σu.


As (X, Y ) is conservative, Theorem 1.4.8 and Theorem 1.5.4 thus imply that(X, Y ) is a semimartingale, whose characteristics (B,C, ν) with respect to thetruncation function (0, χY ) are given by

〈Bt, (u, v)〉 =

∫ t

0

tr (u (bX +BX(Xs))) ds

−∫ t

0

v>(

1

2diag(aY ) +

1

2diag(g>Xsg)

)ds

−∫ t

0

∫v>(eξY − 1− χY (ξY )

)m(dξ)ds

−∫ t

0

∫v>(eξY − 1− χY (ξY )

)M(Xs, dξ)ds,

〈(u, v), Ct(u, v)〉 =

∫ t

0

(v>aY v + 4 tr(uΣ>ΣuXs) + tr(gvv>g>Xs)

+ 2 tr(uΣ>ρv>g>Xs) + 2 tr(gvρ>ΣuXs))ds, (5.47)

ν(dt, dξ) = (m(dξ) +M(Xt, dξ))dt.

Here, 〈(u, v), (x, y)〉 = tr(ux) + v>y, diag(a) is the vector consisting of thediagonal elements of a matrix a, 1 is the vector whose entries are all equal to1 and eξY = (eξY,1 , . . . , eξY,d)>. The canonical semimartingale representation(see Jacod and Shiryaev [2003, Theorem II.2.34]) is thus given by(

Xt

Yt

)=

(xy

)+

(BXt

BYt

)+

(Xct

Y ct

)+

∫ t

0

∫D

(0

χY (ξX)

)(µ(X,Y )(ds, dξ)− ν(ds, dξ))

+

∫ t

0

∫D

(ξX

ξY − χY (ξY )

)µ(X,Y )(ds, dξ),

where (Xc, Y c) denotes the continuous martingale part. From Theorem 3.7.2it follows that there exists a d × d matrix of Brownian motions W such thatXc can be written as

Xct =

∫ t

0

√XsdWsΣ + Σ>dW>

s

√Xs.

In particular,

d〈Xcij, X

ckl〉t = Xt,ik(Σ

>Σ)jl +Xt,il(Σ>Σ)jk +Xt,jk(Σ

>Σ)il +Xt,jl(Σ>Σ)ik.


Thus it remains to establish the representation

Y ct =

∫ t

0

(√aY dZs + g>

√XsdVs

)=: Lt.

To this end, note that the quadratic variation of the right hand side satisfies

d〈Li, Lj〉t = (aY + g>Xtg)ij

and

d〈Xcij, Lk〉t = (g>Xt)ki(Σ

>ρ)j + (g>Xt)kj(Σ>ρ)i.

On the other hand, we can write Ct defined in (5.47) as an (d2 + d)× (d2 + d)matrix (

CXt CXY

t

(CXYt )> CY

t

),

where the entries are given by

CXt,(ij)(kl) = d〈Xc

ij, Xckl〉t

= Xt,ik(Σ>Σ)jl +Xt,il(Σ

>Σ)jk +Xt,jk(Σ>Σ)il +Xt,jl(Σ

>Σ)ik,

CXYt,(ij)k = d〈Xc

ij, Lk〉t= (g>Xt)ki(Σ

>ρ)j + (g>Xt)kj(Σ>ρ)i,

CYt,ij = d〈Li, Lj〉t

= (aY + g>Xtg)ij,

which proves the assertion.

Remark 5.4.6. (i) Note that the only restriction that we impose in Theo-rem 5.4.5 on the parameters, appearing in the equations (5.8) and (5.9)for F and R, is to require that QY (v) = G(vv>), where G is a congruencemap, i.e., an endomorphism of S+

d of the particular form G(a) = gag>

for some d × d matrix g. Indeed, for d > 2, it is an unsolved problemto characterize all linear operators mapping S+

d into S+d but not onto

(see, e.g., Li and Pierce [2001], Schneider [1965]). In the case d = 3,a very simple example of such an endomorphism which cannot be rep-resented as a sum of congruence maps is given in Choi [1975, Theorem2]. Note however that all automorphisms of S+

d are congruence mapswith g an invertible d × d matrix. Since aY + gXg> corresponds to thequadratic covariation of Y , a restriction to automorphisms means thatthe volatility depends on all components of X.

5.5. Examples 163

(ii) From the above semimartingale representation it is easily seen that thecovariation process does not depend on Y . In particular, it is an affineprocess (in its own filtration) on S+

d . Theorem 3.7.1 and Theorem 3.7.2thus imply existence and uniqueness of X. This then also yields existenceand uniqueness of the log-price process, since the dynamics of Y onlydepend on X.

5.5 Examples

In this section we study two particular multivariate affine stochastic volatilitymodels which have been considered in the literature. These examples are spe-cial cases of the model described in Theorem 5.4.5. This result now allows usto read off the necessary conditions for a particular model, like for example thecorrelation structure between the Brownian motions or conditions on the driftparameter. We start with an analog of the Heston model in the multivariatesetting.

5.5.1 Multivariate Heston Model

The dynamics of the multivariate Heston models read as follows,

dXt = (bX +BX(Xt))dt+√XtdWtΣ + Σ>dW>

t

√Xt, (5.48)

dYt = −1

2diag(Xt)dt+

√XtdVt,

where the parameters satisfy the conditions of Theorem 5.2.1 and 5.3.5. Inthe terminology of Theorem 5.2.1, αX = Σ>Σ and QY (v) = vv>. Moreover,

W and V are Brownian motions as specified in Theorem 5.4.5. A version ofthis model was introduced in Da Fonseca et al. [2007b] under more restrictiveparameter assumptions. Indeed, therein bX = kΣ>Σ with k > d − 1 andBX(Xt) = MXt +XtM

> for some d× d matrix M . The correlation structure

between V and W is the same. The restriction bX = kΣ>Σ often leads toproblems when calibrating to data, since the values obtained for k by anoptimization procedure are often smaller than d − 1, which however meansthat the process X does not exist. This isssue can be overcome by allowingfor a more general constant drift.

When assuming the particular linear drift BX(Xt) = MXt + XtM>, the


solutions of the associated Riccati equations

∂Φ(t, u, v)

∂t= Φ(t, u, v) tr(bXΨ(t, u, v)),

∂Ψ(t, u, v)

∂t= 2Ψ(t, u, v)Σ>ΣΨ(t, u, v) +

1

2v>v

+ Ψ(t, u, v)(Σ>ρv> +M) + (vρ>Σ +M>)Ψ(t, u, v)

− 1

2

d∑i=1

vi(eie>i ),

are given by

Ψ(t, u, v) = (uΨ12(t, v) + Ψ22(t, v))−1(uΨ11(t, v) + Ψ21(t, v)),

Φ(t, u, v) = exp

(∫ t

0

tr(bXΨ(s, u, v))ds

),

where(Ψ11(t, v) Ψ12(t, v)Ψ21(t, v) Ψ22(t, v)

)= exp t

(Σ>ρv> +M −2Σ>Σ

12

(vv> −

∑di=1 vi(eie

>i ))−(vρ>Σ +M>)

),

(see Da Fonseca et al. [2007b, Proposition 3.3] for details).As shown in Theorem 5.4.5, the Heston model can be extended by incor-

porating jumps, both in the covariation and the log-price process. This thenyields generalizations of the one-dimensional so-called Bates model.

5.5.2 Multivariate Barndorff-Nielsen-Shepard Model

A multivariate extension of the Barndorff-Nielsen-Shepard model was intro-duced in Pigorsch and Stelzer [2009a] under the name multivariate stochasticvolatility model of OU-type. This model has been analyzed in Barndorff-Nielsen and Stelzer [2011], while calibration related results can be foundin Muhle-Karbe et al. [2010].

Here, the covariation process and the log-price process are described by

dXt = (MXt +XtM>)dt+ dLt, (5.49)

dYt = −1

2diag(Xt)dt−

∫D

(eξY − 1)m(dξ) +√XtdVt + P (dLt),

where M ∈ Rd×d and P : Rd×d → Rd some linear map. Moreover, V is an Rd-valued Brownian motion and L denotes a drift-less matrix subordinator (see,

5.5. Examples 165

e.g., Barndorff-Nielsen and Stelzer [2007]), which is independent of V . Itstriplet is given by (0, 0, U∗m(dξ)), where U denotes the projection of S+

d ×Rd

on S+d and m corresponds to the Levy measure of (L, P (L)). We write eξY for

(eξY ,1, . . . , eξY ,d)> and 1 stands for the vector whose entries are all equal to 1.Here, the truncation function χY can assumed to be 0.

The covariation process X thus follows a positive semidefinite processof OU-type, which was introduced in Barndorff-Nielsen and Stelzer [2007].Note that, as before, the linear drift is specified to be of the particular formBX(Xt) = MXt + XtM

>, which allows to keep analytic tractability whensolving the associated Riccati equations, given by

∂Φ(t, u, v)

∂t= Φ(t, u, v)

(−∫D

v>(eξY − 1)m(dξ)

+

∫D

(etr(ξXΨ(t,u,v))+v>ξY − 1

)m(dξ)

),

∂Ψ(t, u, v)

∂t=

1

2v>v + Ψ(t, u, v)M +M>Ψ(t, u, v)− 1

2

d∑i=1

vi(eie>i ).

Observe that the equation for Ψ is linear and can thus be solved explicitly,while the solution of Φ can be obtained via integration (see, e.g., Muhle-Karbeet al. [2010, Theorem 2.5]).

The relation to the one dimensional Barndorff-Nielsen-Shepard model be-comes clear when analyzing the marginal dynamics of the one dimensionallog-prices. By Muhle-Karbe et al. [2010, Theorem 2.3] (see also Barndorff-Nielsen and Stelzer [2011, Proposition 4.3]), we have

(Yt,i)fidi=

(∫ t

0

−1

2Xs,iids− t

∫D

(eξY,i − 1)m(dξ)

+

∫ t

0

(√Xs,iidZs,i + Pi(dLs)

)),

wherefidi= denotes equality for all finite dimensional distributions. If M is

assumed to be diagonal and P : x 7→ (p1x11, . . . , pdxdd) for some vector p ∈ Rd,then the model for the ith asset is equivalent in distribution to a univariateBarndorff-Nielsen-Shepard model, since every diagonal element of the matrixsubordinator L is obviously a univariate subordinator.

Chapter 6

Applications of PolynomialProcesses

The applications of polynomial processes range from statistical GMM estima-tion procedures to techniques for option pricing and hedging. For instance, theefficient and easy computation of moments can be used for variance reductiontechniques in Monte-Carlo methods.

As we have seen in Section 4, polynomial processes have the property thatthe expressions for all finite moments are analytically known (up to a matrixexponential). In other words, there is a large subset of claims where theprices and hedging ratios (Greeks) are explicitly known. Large can be madeprecise in the following sense: if the law of XT , say µ, is characterized by itsmoments, then “large” means dense, i.e., polynomial claims are dense (withrespect to the L1(µ) norm) in the set of “all” claims. If this is not the case,the payoff function can at least be uniformly approximated by polynomials onsome interval, which can be chosen according to the support of the probabilitydistribution. The explicit knowledge of prices of polynomial claims then allowsto apply variance reduction techniques for Monte-Carlo computations.

6.1 Moment Calculation

By Theorem 4.1.8 we know that there exists a linear map A such that momentsof m-polynomial processes can simply be calculated by computing etA. Indeed,by choosing a basis 〈e1, . . . , eN〉 of Pol≤m(S) the matrix corresponding to thislinear map, which we also denote by A = (Akl)k,l=1,...,N , can be obtained

through Aek =∑N

l=1Aklel. Writing f as∑N

k=1 αkek, we then have

Ptf = (α1, . . . , αN)etA(e1, . . . , eN)′, (6.1)

167

168 Chapter 6. Applications of Polynomial Processes

which means that moments of polynomial processes can be evaluated simplyby computing matrix exponentials.

By means of the one-dimensional Cox-Ingersoll-Ross process

dXt = (b+ βXt)dt+ σ√XtdWt, b, σ ∈ R+, β ∈ R,

we exemplify how moments of order m can be calculated. Its (extended)infinitesimal generator generator is given by

Af(x) =1

2σ2x

d2f(x)

dx2+ (b+ βx)

df(x)

dx.

Applying A to (x0, x1, . . . , xm) yields the following (m+ 1)× (m+ 1) matrix

A =

0 . . .b β 0 . . .0 2b+ σ2 2β 0 . . .0 0 3b+ 3σ2 3β 0 . . .

. . .

0 . . . mb+ m(m−1)2

σ2 mβ

.

Hence Ex[(Xt)

k]

= Ptxk = (0, . . . , 1, . . . , 0)etA(x0, . . . , xk, . . . , xm)′.

Remark 6.1.1. (i) Note that A is a lower triangular matrix, whose eigen-values are the diagonal elements. Since in this case they are all distinct,the matrix is diagonalizable. Of course, there are many efficient algo-rithms to evaluate such matrix exponentials (see, e.g., Golub and VanLoan [1996], Moler and Van Loan [1978]).

(ii) For n > 1, for example in the case of the Heston model, similar closedform expressions can be obtained.

6.1.1 Generalized Method of Moments

In view of this fast technique of moment calculation for polynomial processes,the Generalized Method of Moments (GMM) qualifies for parameter estima-tion and thus also for model calibration. The implementation of a typicalmoment condition of the type

f(Xt, θ) =

Xn1t X

m1t+s − Ex[Xn1

t Xm1t+s]

...Xnqt X

mqt+s − Ex[X

nqt X

mqt+s]

, ni,mi ∈ N, 1 ≤ i ≤ q,

6.2. Pricing - Variance Reduction 169

where θ is the set of parameters to be estimated, is simple since Ex[Xnt X

mt+s

]=

Ex [Xnt EXt [Xm

s ]] can also be computed easily. In the case of one-dimensionaljump-diffusions, Zhou [2003] has already used this method for GMM estima-tion.

6.2 Pricing - Variance Reduction

The fact that moments of polynomial processes are analytically known alsogives rise to efficient techniques for pricing and hedging.

Let X be an m-polynomial process and G : S → Rn a deterministic bi-measurable map such that the discounted price processes are given throughSt = G(Xt) under a martingale measure. Typically G = exp if X are log-prices. We denote by F = φ(ST ) a bounded measurable European claim forsome maturity T > 0, whose (discounted) price at t ≥ 0 is given by the riskneutral valuation formula

pFt = Ex[φ(ST )

∣∣Ft] = EXt [(φ G)(XT )] .

Obviously, claims of the form

F = f G−1(ST ) (6.2)

for f ∈ Pol≤m(S) are analytically tractable, since we have

pFt = Ex[(f G−1)(ST )

∣∣Ft] = PT−tf(G−1(St)) = e(T−t)Af(G−1(St))

for 0 ≤ t ≤ T , where A is the previously defined linear map on Pol≤m(S).Although claims are in practice not of form (6.2), the explicit knowledge of theprice of polynomial claims can be used for variance reduction techniques basedon control variates. Instead of using the estimator πF0 = 1

L

∑Li=1(φ G)(X i

T )in a Monte-Carlo simulation, where X1

T , . . . , XLT are L samples of XT , we can

use

πF0 =1

L

L∑i=1

((φ G)(X i

T )−(f(X i

T )− Ex [f(XT )])),

where f ∈ Pol≤m(S) is an approximation of φG and serves as control variate.Both estimators are unbiased and the second clearly outperforms the first sinceVar

(πF0)< Var

(πF0), where the ratio of the variances depends on the accuracy

of the polynomial approximation.It is worth mentioning that the previous pricing algorithm has also im-

portant consequences for hedging, since the Greeks for “polynomial claims”F = f(XT ) can also be calculated explicitly and efficiently: The coefficients


of the polynomial x 7→ Ex[f(XT )] can be computed using matrix exponentia-tion, taking derivatives of this polynomial is then a simple algebraic operation.To be more precise, the sensitivities of the price process with respect to thefactors X can be calculated by

∇pFt = ∇PT−tf(G−1(St))∇G−1(St).

Assuming a complete market situation, the claim φ(ST ) = φ G(XT ) can bereplicated by a trading strategy η, i.e.,

φ(ST ) = Ex[φ(ST )] +

∫ T

0

ηtdSt.

Similarly, the polynomial claim f(Xt) can be replicated by the delta-hedgingstrategy ∇pFt and we conclude that

φ(ST )− f(XT ) = Ex[φ(ST )]− Ex[f(XT )] +

∫ T

0

(η −∇pFt )dSt.

If we assume that φ(ST )− f(XT ) has small variance, then also the stochasticintegral representing the difference of the cumulative gains and losses of thetwo hedging portfolios, namely the one built by the unknown strategy η andthe one built by the known strategy ∇pFt , is small.

Concerning the approximation of φ G by a polynomial, let us considerthe case St = G(Xt,1) with G : R → R+, meaning that we only have oneasset which depends on the first component of the polynomial process X as itis usually the case in stochastic volatility models. If the Hamburger momentproblem (see, e.g., Reed and Simon [1975]) for the law of XT,1, say µ, admits aunique solution, then the set of all polynomials is dense in L2(µ) (see Akhiezer[1965, Theorem 2.3.3]) and hence also in L1(µ). A sufficient conditions for theuniqueness of a solution to this moment problem is∣∣Ex [Xk

T,1

]∣∣ ≤ Ck!

Rk(6.3)

for some constants C > 0 and R > 0 (see Reed and Simon [1975, ExampleX.4]). This condition can be assured by the existence of exponential momentsof XT,1 around 0, that is, the moment generating function Ex[euXT,1 ] is finitefor all u ∈ (−ε, ε). Indeed, since u 7→ Ex[euXT,1 ] is real analytic on (−ε, ε),there exist an open interval J ⊆ (−ε, ε) and constants C > 0 and R > 0 suchthat for all u ∈ J ∣∣∣∣dkEx[euXT,1 ]

duk

∣∣∣∣ ≤ Ck!

Rk

6.2. Pricing - Variance Reduction 171

(see, e.g., Krantz and Parks [2002, Corollary 1.2.9]). Hence in particular (6.3)is satisfied.

In order to present the positive impact of our variance reduction methodgraphically, we implemented the following affine stochastic volatility model,which was initially proposed by Bates [2000]. The price process is specified asSt = S0e

Xt with dynamics

d

(Xt

Vt

)=

(r − Vt

2− λVt

∫R(eξ − 1)F (dξ)

b− βVt

)dt

+

( √Vt 0

σρ√Vt σ

√1− ρ2

√Vt

)(dBt,1

dBt,2

)+

(dZt0

),

where B is a 2-dimensional Brownian motion and Z a pure jump processin R with jump intensity λv and exponentially distributed jump sizes, i.e.,

F (ξ) = 1ce−

ξc , for some parameter c ∈ R+. Figure 6.1 illustrates the compar-

ison between the Monte-Carlo simulation for European Call prices with andwithout variance reduction. In this example we use the following parameters:

S0 V0 Strike r b β σ ρ λ c

10 0.1 9 0.04 0.08 0.7 0.03 0 1.5 0.05

The polynomial which we take to approximate the payoff function is of degree10 and is chosen such as to minimize the approximation error with respectto the supremum norm in a certain interval (depending on the support ofthe probability distribution). Concerning computation time, we remark thatbeside the one-time calculation of the matrix exponential, the only additionalcomputational effort resulting from the use of the control variates is the eval-uation of a polynomial in each loop. In our MATLAB code, this causes anincrease of computation time of less than 50% per replication. Observing thatone can achieve the same accuracy by using 100 times less replications throughthe polynomial control variates, the computation time (in our MATLAB im-plementation) can be decreased by a factor of more than 65.

Let us finally remark that our variance reduction technique is of particularinterest for affine models for which the generalized Riccati ODEs (see Duffieet al. [2003]) determining the characteristic function cannot be explicitlysolved. Moreover, it can also be applied to derivatives involving several assets,provided that their dynamics are described by a polynomial process. This cansimply be done by approximating European payoff functions depending onseveral variables with multivariate polynomials.


Figure 6.1: Comparison: Monte-Carlo simulation for European Call priceswith and without variance reduction.

Appendix A

Symmetric Cones andEuclidean Jordan Algebras

A.1 Important Definitions

In this section (V, 〈·, ·〉) denotes an Euclidean Jordan algebra and K the cor-responding symmetric cone of squares, as introduced in Definition 3.1.3, Def-inition 3.1.1 and Theorem 3.1.6. The aim of this appendix is to review someimportant notions and results related to Euclidean Jordan algebras.

A.1.1 Determinant, Trace and Inverse

We denote by R[λ] the polynomial ring over R in a single variable λ and, forx ∈ V , we define R[x] := p(x) | p ∈ R[λ]. Then we have

R[x] = R[λ]/J (x)

with the ideal J (x) := p ∈ R[λ] | p(x) = 0. Since R[λ] is a principalring, J (x) is generated by a polynomial, which is referred to as the minimalpolynomial if the leading coefficient is 1. Its degree is denoted by m(x). Wehave

m(x) = mink > 0 | e, x, x2, . . . , xk are linearly dependent

.

Furthermore, the rank of V is the number

r := maxx∈V

m(x), (A.1)

which is bounded by n = dim(V ). An element x is said to be regular ifm(x) = r. By Faraut and Koranyi [1994, Proposition II.2.1], there exist

173

174 Appendix A. Symmetric Cones and Euclidean Jordan Algebras

unique polynomials a1, . . . , ar on V such that the minimal polynomial of everyregular element x is given by

f(λ;x) = λr − a1(x)λr−1 + a2(x)λr−2 + · · ·+ (−1)rar(x).

Using this fact, one introduces, for any x ∈ V , the determinant det(x) andtrace tr(x) as

det(x) := ar(x) and tr(x) := a1(x).

Remark A.1.1. In order to distinguish between elements of V and linearmaps on V , we use the notations Tr(A) and Det(A) for A ∈ L(V ).

An element x is said to be invertible if there exists an element u ∈ R[x]such that x y = e. Since R[x] is associative, y is unique. It is called theinverse of x and is denoted by y = x−1.

Remark A.1.2. We remark that the notions “rank”, “trace” and “determi-nant” are motivated by the fact that for the Euclidean Jordan algebra of realr × r symmetric matrices, the rank is equal to r, and ar(x) and a1(x) are theusual determinant and trace, respectively. The inverse is also the usual one.

Note that the determinant is not multiplicative in general, but we havethe following.

Proposition A.1.3. For all z ∈ V and x, y ∈ R[z],

det(x y) = det(x) det(y).

Moreover, det(e) = 1 and tr(e) = r. In particular,

det(x) det(x−1) = det(x x−1) = 1.

A.1.2 Idempotents, Spectral and Peirce Decomposition

An element p of V is called idempotent if p2 = p, and two idempotents p, q arecalled orthogonal if pq = 0. It is important to note that for idempotents thisnotion of orthogonality coincides with the notion of orthogonality with respectto the scalar product 〈·, ·〉 on V . Finally, a non-zero idempotent element iscalled primitive if it cannot be expressed as a sum of non-zero orthogonalidempotents.

The following results are cornerstones of the theory of Jordan algebras:

A.1. Important Definitions 175

Spectral Decomposition A set of mutually orthogonal primitive idempo-tents p1, . . . , pr such that p1 + · · · + pr = e is called a Jordan frame ofrank r corresponding to the rank of the Euclidean Jordan algebra, asdefined in (A.1). The spectral decomposition theorem (see Faraut andKoranyi [1994, Theorem III.1.2]) states that, for every x ∈ V , thereexists a Jordan frame p1, . . . , pr and real numbers λ1, . . . , λr, such that

x =r∑

k=1

λkpk,

where the numbers λk are uniquely determined by x. Moreover, x isan element of the symmetric cone K if and only if λk ≥ 0 for allk ∈ 1, . . . , r. Again motivated by symmetric matrices, λ1, . . . , λr arereferred to as eigenvalues.

Peirce decomposition 1 Let c be an idempotent in V . Define, for k =0, 1/2, 1 the subspaces V (c, k) := x ∈ V : c x = kx. Then, by Farautand Koranyi [1994, Proposition IV.1.1], V can be written as the directorthogonal sum

V = V (c, 1)⊕ V (c, 1/2)⊕ V (c, 0).

Moreover, any x ∈ V decomposes with respect to this decompositioninto

x = x1 + x 12

+ x0, (A.2)

where xk ∈ V (c, k) for k = 0, 1/2, 1.

By Faraut and Koranyi [1994, Proposition IV.1.1], the Peirce spacesV (c, k) satisfy certain so called Peirce multiplication rules:

V (c, 1) V (c, 0) = 0, (A.3)

(V (c, 1) + V (c, 0)) V (c, 1/2) ⊂ V (c, 1/2) , (A.4)

V (c, 1/2) V (c, 1/2) ⊂ V (c, 1) + V (c, 0), (A.5)

P (V (c, 1/2))V (c, 1) ⊂ V (c, 0). (A.6)

Peirce decomposition 2 Let p1, . . . , pr be a Jordan frame. Then V can bewritten as the direct orthogonal sum

V =⊕i≤j

Vij, (A.7)


where Vii = V (pi, 1) = Rpi and Vij = V (pi, 1/2)∩V (pj, 1/2) (see Farautand Koranyi [1994, Theorem IV.2.1]).

Moreover, the projection onto Vii is given by the quadratic representationP (pi), and the projection onto Vij by 4L(pi)L(pj).

If V is simple, then the dimension of Vij, i < j, denoted by d = dimVij,is independent of i, j and the Jordan frame. It is called Peirce invariant.If V is simple of dimension n and rank r, then we have by Faraut andKoranyi [1994, Corollary IV.2.6]

n = r +d

2r(r − 1).

Corresponding to the decomposition (A.7), we can write for all x ∈ V

x =r∑i=1

xi +∑i<j

xij,

with xi ∈ Rpi and xij ∈ Vij. One can think of x as a symmetric r ×r matrix, whose diagonal elements are the xi and whose off-diagonalelements are the xij.

Example A.1.4. (i) Consider V = Rn equipped with the Euclidean scalarproduct. Together with the Jordan product

x y = (x1y1, . . . , xnyn)

(element-wise multiplication in R), V is a Jordan algebra and the cor-responding reducible cone is Rn

+. The idempotents are vectors consistingonly of zeros and ones. The non-zero primitive idempotents are the unitvectors. The spectral decomposition (λ1, . . . , λn) are simply the coordi-nates of x in the Cartesian coordinate system.

(ii) Consider V = Sr, the space of real symmetric r × r-matrices. Here,the idempotents correspond to the orthogonal projections and the non-zero primitive idempotents are the orthogonal projections on one-dimens-ional subspaces. In the spectral decomposition, λ1, . . . , λr are the usualeigenvalues of x and p1, . . . , pr are the orthogonal projections on thecorresponding eigenvectors.

Concerning Peirce decomposition 1, the matrix of block form(Ik 00 0

)

A.1. Important Definitions 177

is an idempotent of V . The associated Peirce decomposition of a sym-metric matrix is then given by(

x1 x>12

x12 x0

)=

(x1 00 0

)+

(0 x>12

x12 0

)+

(0 00 x0

).

As already established above, Peirce decomposition 2 corresponds to

x =r∑i=1

. . .. . .

xi. . .

. . .

+∑i<j

...

xij...

xij...

.

Note that the dimension d of Vij, i < j, is 1 here.

A.1.3 Classification of Simple Euclidean Jordan Alge-bras

For sake of completeness, we here state the classification of all simple EuclideanJordan algebras and their corresponding irreducible cones. This classificationis summarized in the following table. Indeed, every simple Euclidean Jordanalgebra is isomorph to one of these cases.

Here, H and O denote the algebra of quaternions and octonions, respec-tively (see Faraut and Koranyi [1994, page 84]). We denote by Herm(r,A) thereal vector space of Hermitian matrices with entries in A, where A correspondseither to C,H or O.

K V dimV rkV d

S+r Sr

12r(r + 1) r 1

Herm+(r,C) Herm(r,C) r2 r 2

Herm+(r,H) Herm(r,H) r(2r − 1) r 4

Lorentz cone R × Rn−1 n 2 n− 2

Exceptional cone Herm(3,O) 27 3 8


A.2 Some Results

In this section we collect a number of lemmas and propositions which are usedin the proofs of C 3. In most cases, we only cite the assertions without proofs,as they can be found in Faraut and Koranyi [1994]. For the sake of notationalconvenience we always assume that V is a simple Euclidean Jordan algebra ofdimension n and rank r, equipped with the natural scalar product

〈·, ·〉 : V × V → R, 〈x, y〉 := tr(x y).

However, the particular form of the scalar product and the assumption thatV is simple is not always needed.

Lemma A.2.1. (i) Let a, b be idempotents in V . Then 〈a, b〉 ≥ 0. More-over, 〈a, b〉 = 0 if and only if a b = 0.

(ii) Let a, b ∈ K and suppose that 〈a, b〉 = 0. Then a b = 0.

Proof. For (i) we follow the proof of Nomura [1993]: Let b = b0 + b 12

+ b1

be the Peirce decomposition of b relative to a. Then 〈a, b〉 = 〈a b, b〉 =12‖b1/2‖2 + ‖b1‖2, from which the assertion follows.

For (ii) we follow the proof of Hertneck [1962]: Let a, b ∈ K and supposethat 〈a, b〉 = 0. Since b ∈ K there exists c ∈ V such that b = c2. Thus

0 = 〈a, b〉 = 〈a, c2〉 = 〈L(a)c, c〉.

But L(a) is positive semidefinite on V , and we conclude that L(a)c = ac = 0.Hence

〈b, a b〉 = 〈c2, a c2〉 = 〈c4, a〉 = 〈c3, a c〉 = 0,

and (ii) follows.

Proposition A.2.2. The following assertions hold true:

(i) An element x is invertible if and only if P (x) is invertible. Then wehave

P (x)x−1 = x,

P (x)−1 = P (x−1).

(ii) If x and y are invertible, then P (x)y is invertible and

(P (x)y)−1 = P (x−1)y−1.

A.2. Some Results 179

(iii) The differential map x 7→ x−1 is −P (x)−1, that is,

d

dt(x−1 + tu)|t=0 = −P (x−1)u.

(iv) det(P (x)y) = (det x)2 det y.

(v) ∇ ln detx = x−1.

Proof. See Faraut and Koranyi [1994, Proposition II.3.1, II.3.3 (i), II.3.3 (ii),III.4.2 (i), III.4.2 (ii)]

Finally, let us introduce the following notation. For x and y in V we write

x y = L(x y) + [L(x), L(y)],

where [L(x), L(y)] = L(x)L(y) − L(y)L(x). The following transformation isused to prove Lemma 3.4.6:

Definition A.2.3. Let c be an idempotent in V . For z in V (c, 12) we define

the Frobenius transformation τ(z) by τ(z) = exp(2 z c).

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