Malliavin calculus and its applications

B.Sc. Thesis

Malliavin calculus and itsapplications

Adam Gyenge

Supervisor: Dr. Tamas SzabadosAssociate professorInstitute of MathematicsDepartment of Stochastics

Budapest University of Technology and Economics

2010

Contents

1 Introduction 11.1 The structure of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Theory of Malliavin calculus 32.1 Ito integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.1 Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1.2 Construction of the Ito integral . . . . . . . . . . . . . . . . . . . . . 4

2.2 Isonormal Gaussian processes . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Wiener chaos expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3.1 Iterated Ito- and multiple stochastic integrals . . . . . . . . . . . . . 82.3.2 Wiener chaos and iterated Ito integrals . . . . . . . . . . . . . . . . 9

2.4 The derivative operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4.1 Closability and other properties of the derivative operator . . . . . . 122.4.2 The derivative in the white noise case . . . . . . . . . . . . . . . . . 14

2.5 The divergence operator and the Skorohod integral . . . . . . . . . . . . . . 162.5.1 The Skorohod integral . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.6 The Clark-Ocone formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3 Applications 223.1 Stochastic differential equations . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.1.1 Absolute continuity of distributions . . . . . . . . . . . . . . . . . . 223.1.2 Lie bracket and Hormander’s condition . . . . . . . . . . . . . . . . 243.1.3 Absolute continuity under Hormander’s condition . . . . . . . . . . . 25

3.2 Relation to Stein’s method . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.2.1 Stein’s lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.2.2 Ornstein-Uhlenbeck operators . . . . . . . . . . . . . . . . . . . . . . 293.2.3 Bounds on the Kolmogorov distance . . . . . . . . . . . . . . . . . . 30

3.3 Financial mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.3.1 Investments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.3.2 Black-Scholes model . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4 Summary and conclusions 36

i

Chapter 1

Introduction

The Malliavin calculus (or the stochastic calculus of variations) is an infinite dimensionaldifferential calculus on the Wiener space. The foundations of the subject were developedin the late 1970’s, mainly in the two seminal works [6] and [7], in order to study theexistence and smoothness of density for the probability laws of random vectors. Theoriginal motivation and the most important application of this theory has been to providea probabilistic proof of Hormander’s hypoellipticity theorem. The theory was furtherdeveloped by Stroock, Bismut, Watanabe and others, and has gained importance in severaldomains of applications far beyond the original problem.

Because of the many different branches of research and huge number of interestingresults in the topic it is not possible to cover everything in this short paper. We focusedrather on the most important concepts and the logical relations of the results. During theproject the following research questions were stated:

1. What are the key concepts and results in the recently developed mathematical fieldof Malliavin calculus?

2. In which application areas has the theory been adapted for use?

1.1 The structure of the thesis

The rest of this thesis is organized as follows:

• Chapter 2 introduces the theory of Malliavin calculus in the framework of isonor-mal Gaussian processes. We tried to collect the most important concepts and theo-rems without going into the technical details. Many theorems are presented withoutproofs, or with only an outline of the proof. Instead, the focus is rather on showingthe importance of the different notions and results, and the relations between them.The chapter follows at some points the excellent monograph of David Nualart [9],that contains all the missing proofs and much more. The notations and the under-lying concepts are usually from functional analysis and stochastic calculus. For areference on these topics we refer to [11] and [5].

1

CHAPTER 1. INTRODUCTION 2

• In the recent years, the theory of Malliavin calculus has been applied in severaldifferent areas. Out of the many important and interesting applications three aresummarized in Chapter 3. This includes the original motivating domain of stochas-tic differential equations, as well as two more recent fields: the connection to Stein’smethod for limit theorems and applications in financial mathematics.

• Finally, in Chapter 4 the thesis is summarized and some directions for futureresearch are proposed.

I am thankful to my supervisor Tamas Szabados, who encouraged me to study stochas-tic calculus and guided and supported me from the beginning. I am also grateful to Pro-fessor Denes Petz, from whom I have learnt functional analysis and who helped me inseveral aspects. Lastly, I offer my regards to all of those who supported me in any respectduring the completion of the project.

Budapest, 10th May 2010

Adam Gyenge

Chapter 2

Theory of Malliavin calculus

This chapter presents the most important theoretical concepts and results of the stochasticcalculus of variations, also known as Malliavin calculus. The introduction is based onthe monograph of Nualart [9], and uses the context of isonormal Gaussian processes.After defining the isonormal Gaussian processes, the Wiener chaos decomposition and thedifferent operators (derivative and divergence) are concerned.

2.1 Ito integral

The Brownian motion, or Wiener process is definitely one of the most important stochasticprocesses and it is the basic model in many areas of stochastic analysis. It is not possible todefine the Lebesgue-Stieltjes integral of a process in all cases with respect to the Brownianpathes or in a more general case, with respect to semimartingales. Instead, the Ito integralextends the methods of calculus to these stochastic processes.

2.1.1 Brownian motion

Definition 2.1. The Brownian motion W (t), t ∈ R+ is a stochastic process with thefollowing defining properties [5]:

1. (Independence of increments) W (t)−W (s) is independent of Fs for all t > s, whereFs is the σ-field generated by the past: Fs = σW (u), u ≤ s.

2. (Normal increments) W (t)−W (s) has a Normal distribution with mean 0 and vari-ance t− s. This implies that W (t) has a N(0, t) distribution.

3. (Continuity of paths) W (t) is almost surely continuous everywhere in t.

Beyond these axioms it is usually also assumed that the process starts at W (0) = 0.The time interval on which the Brownian motion is defined is [0, T ] for some T > 0, whichis allowed to be infinite. It can be shown that such a process exist.

A slightly different approach is the canonical space of the Brownian motion, where it isassumed that continuous functions are the possible outcomes for the trajectories, and we

3

CHAPTER 2. THEORY OF MALLIAVIN CALCULUS 4

define the process as a probability space. This means that the probability space is definedon Ω = C0([0, T ]), the set of continuous functions from [0, T ] to R starting at 0 (thatis called the classical Wiener space). On this set a topology is induced by the uniformnorm: ‖f‖ = supx∈[0,T ] f(x), and F , the σ-field of measurable sets is the Borel σ-field ofthis topology, that is, the smallest σ-field containing all the open sets. The probabilitymeasure P (or the law of the process) is the Wiener measure defined on the cylinder setsof Ω:

P (ω;ω(t1) ∈ F1, . . . , ω(tk) ∈ Fk) = P (W (t1) ∈ F1, . . . ,W (tk) ∈ Fk) =

=∫F1×···×Fk

ρ(t1, 0, x1)ρ(t2 − t1, x1, x2) · · · ρ(tk − tk−1, xk−1, xk)dx1 · · · dxk ,

where Fi ∈ B(R), 0 ≤ t1 < · · · < tk and

ρ(t, x, y) = (2πt)−1/2 exp−|x− y|2

2t.

Definition 2.2. The quadratic variation of a stochastic process X(t) on the time interval[0, t] is defined as

[X](t) = limn→∞

n∑i=1

|X(tni )−X(tni−1)|2 ,

if the sum converges stochastically, where the limit is taken over all shrinking partitionsof [0, t], with δn = maxi(tni+1 − tni )→ 0, as n→ 0.

In the case of Brownian motion, the sum above converges in L2 to [W ](t) = t, andsince the paths are continuous, the total variation of this process is infinite. Therefore,it is not possible to define the usual Lebesgue-Stieltjes integral with respect to Brownianmotion for processes of the same kind. The Ito integral overcomes this issue.

2.1.2 Construction of the Ito integral

Consider the Brownian motion on an interval [0, T ] ⊆ R+, that can be also infinite.Let Ft, 0 ≤ t ≤ T denote the natural filtration of the Brownian motion, L2(Ω) =L2(Ω,F , P ), and L2([0, T ] × Ω) = L2([0, T ] × Ω,B([0, T ]) ⊗ F , λ × P ), the set of squareintegrable stochastic processes on [0, T ]. From here, t is always in [0, T ].

Definition 2.3. A stochastic process u(t) is adapted, if u(t) is Ft measurable for all t.L2a([0, T ]×Ω) ⊂ L2([0, T ]×Ω) is the subspace of square-integrable, adapted processes. A

process is an elementary adapted process, if there exists a partition 0 ≤ t1 < · · · < tn+1, anda set of random variables F1, . . . , Fn, such that every Fi is an Fti measurable and squareintegrable random variable for all i, and u(t) =

∑ni=1 Fi1(ti,ti+1](t). E ⊂ L2

a([0, T ] × Ω) isthe set of elementary adapted processes.

Definition 2.4. The Ito integral (or stochastic integral) of an elementary adapted processu(t) =

∑ni=1 Fi1(ti,ti+1](t) with respect to the Brownian motion W (t) is∫ T

0u(t)dW (t) =

n∑i=1

Fi(W (ti+1)−W (ti)) (2.1)


The integral of an elementary adapted process is a function in L2(Ω), that is, a squareintegrable random variable. The integral is linear, and has the following properties:

E(∫ T

0u(t)dW (t)) = 0 (2.2)

E(|∫ T

0u(t)dW (t)|2) = E(

∫ T

0u(t)2dt) (2.3)

From (2.3) it follows that the linear operator u(t) 7→∫ T

0 u(t)dW (t) is an E → L2(Ω)isometry. It can be shown that E is a dense subspace of L2

a([0, T ] × Ω). This allows toextend the Ito integral to the whole L2

a([0, T ] × Ω), and for this extension the propertiesabove still hold.

With more advanced techniques it is possible to extend the Ito integral to the class ofmeasurable and adapted processes such that

∫ T0 u(t)2dt < ∞ almost surely. In this case,

the process∫ t

0 u(s)dW (s) is a continuous local martingale, while in the specific case whenu(t) ∈ L2

a([0, T ]× Ω), then the integral process is a continuous martingale.

2.2 Isonormal Gaussian processes

Definition 2.5. Let (Ω,F , P ) be a complete probability space. A subspace G ⊆L2(Ω,F , P ) is a centered Gaussian family, if it is closed, and all the elements of G areGaussian random variables with zero mean.

Let H be a real separable Hilbert space, with the inner product denoted as 〈 , 〉.

Definition 2.6. A centered Gaussian family W ⊆ L2(Ω,F , P ) is an isonormal Gaussianprocess on H, if W = W (h), h ∈ H, that is, W is parametrized by the elements of H,such that E(W (h)W (g)) = 〈g, h〉 for all h, g ∈ H.

Remark. It can be shown that the mapping h 7→W (h) is linear and by the definition it isan isometry.

The isonormal Gaussian processes were defined by Dudley in [2]. It has turned outthat this general notion allows the treatment of many abstract Gaussian families. Thefollowing three examples will be very important in the thesis.

Example 2.1. Let H = Rd for some integer d ≥ 1, and let (e1, . . . , ed) be an orthonormalbasis in Rd with respect to the standard Euclidean inner product. Let (Z1, . . . , Zd) be a setof independent and identically distributed random variables, where each of the componentsare distributed as N(0, 1). For every h ∈ H take the decomposition with respect to thebasis: h =

∑dj=1 cjej , and set W (h) =

∑dj=1 cjZj . Then W = W (h), h ∈ Rd is an

isonormal Gaussian process over Rd.

Example 2.2. Consider the special case, when the underlying Hilbert space H isL2([0, T ],B, µ), where µ is a σ-finite measure without atoms. Then, E(W (1A)W (1B)) =∫ T

0 1A1Bdµ(t) =∫ T

0 1A∩Bdµ(t) = 0, for all disjoint A,B ∈ B. By standard measure


theoretic approximating arguments it can be shown that the set W (A) = W (1A), A ∈B, µ(A) <∞ characterizes W . The function W : B → L2(Ω,F , P ), A 7→W (A) = W (1A)is an L2(Ω) valued Gaussian random measure on ([0, T ],B). This means that

1. For all A ∈ B, the random variable W (A) has the distribution N(0, µ(A))

2. If A1, . . . , An are disjoint sets, the W (A1), . . . ,W (An) are independent

This is called the white noise measure (or Brownian measure) based on µ.If µ is the Lebesgue measure on [0, T ], then the continous version of the process W (t) =

W (1[0,t)) is the standard Brownian motion, and W (h) =∫ T

0 h(t)dW (t) is the Ito integralwith respect to W (t) for all h ∈ L2([0, T ]).

Example 2.3. Similarly to the one dimensional case, it is possible that H = L2([0, T ]×1, . . . , d) ∼= L2([0, T ]; Rd). In this case the continuous version of W (t) = (Wi(t))di=1 =(W ([0, t] × i))di=1, t ∈ [0, T ] is a d-dimensional Brownian motion, and W (h) =∑d

i=1

∫ T0 hi(t)dWi(t) is the multidimensional Ito integral with respect to W (t).

2.3 Wiener chaos expansion

For n ≥ 1 the n-th Hermite polynomial is defined as

Hn(x) =(−1)n

n!e

x2

2dn

dxne−

x2

2 , (2.4)

and let H0(x) = 1. An interesting property of the Hermite polynomials is that Hn(−x) =(−1)nHn(x), for all n ≥ 1. Another important property stated in the next Lemma isconnected to orthogonality in L2(Ω).

Lemma 2.7. If X and Y are two random variables having standard normal distribution,then for all n,m ≥ 1

E(Hn(X)Hm(Y )) =

0 if n 6= m .

1n!(E(XY ))n if n = m .

For an arbitrary random variable X : Ω → R the set FX = X−1(B), B ∈ B ⊆ Ωis a σ-field, that is called the generated σ-field by X. Similarly, the generated σ-field bya set of random variables can be defined. Let G be the σ-field generated by the randomvariables W = W (h), h ∈ H, where W is an isonormal Gaussian process.

Lemma 2.8. The variables W = eW (h), h ∈ H form a complete subset of L2(Ω,G, P ).In other words, if an X ∈ L2(Ω,G, P ) is orthogonal to all eW (h), then X = 0 almost surely.


Proof. Let X be a random variable orthogonal to all such variables: E(XeW (h)) = 0 forall h ∈ H. Since h→W (h) is linear,

E(X expm∑i=1

tiW (hi)) = 0 (2.5)

for any t1, . . . , tm ∈ R and h1, . . . , hm ∈ H. For a fixed set h1, . . . , hm ∈ H and m ≥ 1,define a measure ν on the Borel sets B of Rm: ν(B) = E(X1B(W (h1), . . . ,W (hm))) =∫X(ω)1B(W (h1, ω), . . . ,W (hm, ω))dP (ω). For example, if m = 1, then

ν((−∞, x]) = E(X1(−∞,x](W (h1)))

=∫X(ω)1(−∞,x](W (h1, ω))dP (ω) =

∫X(ω)1(W (h1,ω)≤x)dP (ω) .

Let us denote by ν the pull-back of ν onto Ω. Taking into account that

dν/dP = X(ω)1((W (h1,ω),...,W (hm,ω))∈B) ,

the moment generating function of ν is

Lν(s) =∫

Rm

e−〈s,x〉dν(x) =∫

Ωe−〈s,(W (h1,ω),...,W (hm,ω))〉dν(ω)

=∫

Ωe−〈s,(W (h1,ω),...,W (hm,ω))〉X(ω)1((W (h1,ω),...,W (hm,ω))∈Rm)dP (ω)

= E(e−〈s,(W (h1),...,W (hm))〉X) , (2.6)

for all s ∈ Rm.It follows form (2.5) and (2.6) that Lν(s) = 0 for all s ∈ R. Hence, this measure is

zero everywhere, which implies that E(X1G) = 0 for all G ∈ G. By the properties of theexpected value, this means that X = 0 almost surely.

Definition 2.9. The n-th Wiener chaos of L2(Ω,F , P ), denoted byHn is the closed linearsubspace, generated by the random variables Hn(W (h)), h ∈ H, ‖h‖ = 1, where Hn isthe n-th Hermite polynomial, defined by (2.4).

The following properties of the n-th Wiener chaos can be proved easily:

1. Since H0(x) = 1, H0 consists of the constants.

2. Since H1(x) = x, H1 = W (h), h ∈ H) = W .

3. By Lemma 2.7, the subspaces Hn and Hm are orthogonal whenever n 6= m.

In addition to the orthogonality, more happens to be true. The following theorem willbe crucial in the next parts.

Theorem 2.10. The Hilbert space L2(Ω,G, P ) can be decomposed into the infinite directsum of the subspaces Hn:

L2(Ω,G, P ) =∞⊕n=0

Hn .


Proof. Let X ∈ L2(Ω,G, P ) such that X is orthogonal to Hn for all n ≥ 0. That is,

E(XY ) = 0, ∀Y ∈ Hn, ∀n ≥ 0 .

By the linearity of the expected value, it is enough to assume that

E(XHn(W (h))) = 0, ∀h ∈ H, ‖h‖ = 1, ∀n ≥ 0 .

It is a known fact that for each n ≥ 0, the polinomial p(x) = xn can be expressed bythe linear combination of the Hermite-polinomials Hr, where 0 ≤ r ≤ n. Again, by thelinearity of the expected value it follows that E(XW (h)n) = 0 for all n ≥ 0. Therefore,E(X exp(tW (h))) = 0 for all t ∈ R and h ∈ H with norm one. Due to Lemma 2.8 thismeans that X = 0 almost everywhere.

Let Jn : L2(Ω,G, P ) → Hn denote the orthogonal projection onto the n-th Wienerchaos. By Theorem 2.10 for all X ∈ L2(Ω,G, P ) we have the following expansion (con-verging in L2(Ω,G, P )) that is called the Wiener chaos expansion:

X =∞∑n=0

Jn(X) .

2.3.1 Iterated Ito- and multiple stochastic integrals

As it was mentioned earlier, in the (one-dimensional) white noise case the underlyingHilbert spaceH is L2([0, T ],B, µ), where µ is a σ-finite measure without atoms. We will de-fine the so-called multiple stochastic integral Im(f) of a function f ∈ L2([0, T ]m,Bm, µm) =(L2([0, T ],B, µ))⊗m for positive integers m. Let Em denote the set of elementary functionsof the form

f(t1, . . . , tm) =m∑

i1,...,im=1

ai1···im1Ai1×···×Aim

(t1, . . . , tm) , (2.7)

where A1, . . . , Am are pairwise-disjoint sets having finite measures and the coefficientsai1···im are zero if any of the two indices i1, . . . , im are equal. This means that f is zero onthe rectangles that intersect any diagonal subspace ti = tj , i 6= j.

Definition 2.11. The multiple stochastic integral of a function of the form (2.7) is therandom variable defined as

Im(f) =m∑

i1,...,im=1

ai1···imW (Ai1) · · ·W (Aim) .

Remark. This integral can be thought of as integral with respect to the m dimensionaltensor power of the white noise “measure”. It is important that the definition does notdepend on the particular representation of f .

Definition 2.12. The symmetrization of an arbitrary function f is

f(t1, . . . , tm) =1m!

∑σ

f(tσ(1), . . . , tσ(m)) ,


where the sum is taken over all permutations σ of (1, . . . ,m). The function f is symmetricif f ≡ f . The (Kolmogorov quotient of the) set of square integrable symmetric functionsover [0, T ]m is denoted as L2([0, T ]m).

It is easy to see that Im(f) is linear. By the definition, if f is of the form f(t1, . . . , tm) =1Ai1

×···×Aim(t1 . . . , tm), then Im(f) = Im(f). From this and the linearity it follows that

Im(f) = Im(f) for all elementary functions (even for non-symmetrical ones) of the form(2.7). The next property gives the inner product of two multiple stochastic integrals:

〈Im(f), In(g)〉L2(Ω) = E(Im(f)In(g)) =

0 if m 6= n .

m!〈f , g〉L2([0,T ]m) if m = n .(2.8)

It can be shown that the space Em is dense in L2([0, T ]m). From (2.8) follows the isometryinequality for In:

‖Im(f)‖2L2(Ω) = E(Im(f)2) = m!‖f‖2L2(Tm) ≤ m!‖f‖2L2(Tm) . (2.9)

Therefore, by a similar argument as for the Ito integral, the multiple stochastic integraloperator Im can be extended to a linear and continuous operator from L2([0, T ]m) toL2(Ω,F , P ), such that the mentioned properties still hold. Symbolically, the multiplestochastic integral can be denoted as

Im(f) =∫

[0,T ]mf(t1, . . . , tm)dW (t1) · · · dW (tm) =

∫[0,T ]m

f(t1, . . . , tm)dW⊗n(t) .

Suppose f ∈ L2([0, T ]m), that is, f is symmetric and square integrable on [0, T ]m.Let Sm = (t1, . . . , tm) ∈ [0, T ]m; 0 ≤ t1 ≤ · · · ≤ tm ≤ T, the m-dimensional simplexthat occupies the 1

m! fraction of the whole m-dimensional box [0, T ]m. Then Im(f) can beexpressed as

Im(f) =∫

[0,T ]mf(t1, . . . , tm)dW⊗n(t)

= m!∫ T

0

∫ tm

0· · ·∫ t2

0f(t1, . . . , tm)dW (t1) · · · dW (tm) , (2.10)

that is an iterated Ito integral of f over Sm. This integral makes sense, because at eachintegration with respect to dW (ti) the integrand is adapted and square integrable withrespect to dP × dti, 1 ≤ i ≤ n. The equality is valid if f is an elementary function of theform (2.7). In the general case it follows from a density argument, taking into account thatsince f ≡ f , the inequality is equality in (2.9), and then the iterated Ito integral satisfiesthe same isometry property as the multiple stochastic integral. Hence, their extensionscoincide.

2.3.2 Wiener chaos and iterated Ito integrals

In the next part it will be shown that there is a strong connection between the Wienerchaos in the white noise case and the iterated Ito integrals. Suppose f ∈ L2([0, T ]p) and


g ∈ L2([0, T ]q). Let f ⊗ g = f(t1, . . . , tp)g(tp+1, tp+q) be the tensor product of f and g.For any 1 ≤ r ≤ min(p, q) its contraction by r indices can be defined by

(f ⊗r g)(t1, . . . , tp+q−2r) =∫

[0,T ]rf(t1, . . . , tp−r, s)g(tp−r+1, . . . , tp+q−2r, s)µr(ds) .

The tensor product and the contractions are not necessarily symmetric even though f andg are symmetric. Their symmetrizations will be denoted by f⊗g and f⊗rg, respectively.

The product of the multiple integrals of two symmetric functions satisfies the followingproperties:

(i) If f ∈ L2([0, T ]p) and g ∈ L2([0, T ]), then

Ip(f)I1(g) = Ip+1(f ⊗ g) + pIp−1(f ⊗1 g) . (2.11)

(ii) If f ∈ L2([0, T ]p) and g ∈ L2([0, T ]q), then

Ip(f)Iq(g) =min(p,q)∑r=1

r!(p

r

)(q

r

)Ip+q−2r(f ⊗r g) . (2.12)

The first property can be proved by approximation with elementary functions. The secondproperty follows from the first one by induction with respect to the index q.

The next result gives the relationship between multiple stochastic integrals and Hermitepolynomials.

Theorem 2.13. Let Hm(x) be the m-th Hermite polynomial and let h ∈ L2([0, T ]) be suchthat ‖h‖ = 1. Then

m!Hm(W (h)) =∫

[0,T ]mh(t1) · · ·h(tm)dW (t1) · · · dW (tm) = Im(h⊗m) . (2.13)

Proof. The theorem can be proved by induction over m. For m = 1 it follows from thedefinition. Let us assume that it holds for 1, . . . ,m. The Hermite polynomials have thefollowing recursion property:

(m+ 1)Hm+1(x) = xHm(x)−Hm−1(x), m ≥ 1 .

Using this and (2.11), we have

Im+1(h⊗(m+1)) = Im(h⊗m)I1(h)−mIm−1

(h⊗(m−1)

∫Th(t)2µ(dt)

)= m!Hm(W (h))W (h)−m(m− 1)!Hm−1(W (h))

= m!(m+ 1)Hm+1(W (h)) = (m+ 1)!Hm+1(W (h)).

Theorem 2.14. The multiple integral Im maps L2([0, T ]m) onto the Wiener chaos Hm.


Proof. By (2.9), the multiple integral of a symmetric function f ∈ L2([0, T ]m) exists andhas a finite variance, which means that the multiple integral can be considered as an iso-metric mapping on L2([0, T ]m) (except for the factor m!). Since L2([0, T ]m) is a closedsubspace of L2([0, T ]m) and Im is continuous and linear, its image Im(L2([0, T ]m)) willbe again closed subspace in L2(Ω). By (2.13), this image contains the random variablesHm(W (h)), h ∈ H, ‖h‖ = 1. Consequently, Hm ⊂ Im(L2([0, T ]m)). Because the orthogo-nality between multiple integrals of different order, the image Im(L2([0, T ]m)) is orthogonalto Hn whenever n 6= m. This means that Im(L2([0, T ]m)). The general case follows fromthe fact that Im(f) = Im(f) for all f ∈ L2([0, T ]m).

This leads to the following reformulation of the Wiener chaos expansion that can beproved by an orthonormal basis expansion of the functions.

Theorem 2.15. For a random variable F ∈ L2(Ω,G, P ), where G is the σ-field generatedby W, there exists a unique set of symmetric functions fn ∈ L2([0, T ]n), n ≥ 0, such that

F =∞∑n=0

In(fn) , (2.14)

where f0 = E(F ), and I0 is the identity mapping on the constants.

We will denote by FA the σ-field generated by the random variables the randomvariables W (B), B ⊂ A,B ∈ B, µ(B) <∞.

Proposition 2.16. Suppose F ∈ L2(Ω,G, P ) with the representation (2.14), and let A ∈B. Then

E(F |FA) =∞∑n=0

In(fn1⊗nA ) .

2.4 The derivative operator

Let C∞p (Rn) denote the set of all arbitrary many times continuously differentiable functionsf : Rn → R, such that f and all of its partial derivatives have polynomial growth. Thepartial derivative of f with respect to the i-th variable will be denoted by ∂if , and ∇f =(∂1f, . . . , ∂nf) is the gradient of f . The set of smooth random variables is defined as

S = F : F = f(W (h1), . . . ,W (hn)) , (2.15)

where f ∈ C∞p (Rn), h1, . . . , hn ∈ H and n ≥ 1. Let S0 ⊂ S be the dense subspace wheref belongs to C∞0 (Rn), that is, f has a compact support. Similarly, P ⊂ S is the densesubspace containing the smooth random variables, where f is a polynomial in n variables.Similarly,

Example 2.4. In the white noise case the random variables in P are called the Wienerpolynomials.


Definition 2.17. The derivative of a smooth random variable F = f(W (h1), . . . ,W (hn))is the random variable DF : Ω→ H given by

DF =n∑i=1

∂if(W (h1), . . . ,W (hn))hi (2.16)

Example 2.5. Consider the simplest case, when f is univariate and f(x) = x. Then, bythe definition DW (h) = Df(W (h)) = h.

We can compute the inner product in H of DF and an arbitrary vector h ∈ H:

〈DF, h〉 = 〈n∑i=1

∂if(W (h1), . . . ,W (hn))hi, h〉

= 〈n∑i=1

(limε→0

1ε

[f(. . . ,W (hi) + ε, . . . )− f(. . . ,W (hi), . . . )])hi, h〉

= limε→0

n∑i=1

1ε

[f(. . . ,W (hi) + ε, . . . )− f(. . . ,W (hi), . . . )]〈hi, h〉 (2.17)

= ∇f |(W (h1),...,W (hn)) · (〈h1, h〉, . . . , 〈hn, h〉)︸︷︷︸∈Rn

= ∇f · (〈h1, h〉, . . . , 〈hn, h〉)|(W (h1),...,W (hn))

= limε→0

1ε

[f(W (h1) + ε〈h1, h〉, . . . ,W (hn) + ε〈hn, h〉)− f(W (h1), . . . ,W (hn))] .

(2.18)

where in the last equation we used the well-known property of the gradient that∇f ·v is justthe directional derivative from v. For each h ∈ H and e > 0 the shifted (or perturbed)Gaussian process can be defined as W h

ε = W (g) + ε〈g, h〉, g ∈ H. Similarly for eachF ∈ S, there is a shifted random variable F hε = f(W (h1)+ ε〈h1, h〉, . . . ,W (hn)+ ε〈hn, h〉),defined on W h

ε . Therefore, by (2.18) DF can be thought as the total derivative of F and〈DF, h〉 = limε→0

1ε [F

hε − F ] as the directional derivative from the direction h.

Definition 2.18. The directional derivative of a smooth random variable F ∈ S from thedirection h ∈ H is DhF = 〈DF, h〉.

2.4.1 Closability and other properties of the derivative operator

Lemma 2.19. If F is a smooth random variable and h ∈ H, then

E(DhF ) = E(FW (h)) .

Proof. By the linearity of the assignment h → W (h), we can assume that ‖h‖ = 1.There exist elements e2, . . . , en in H, such that h, e2, . . . , en is an orthonormal system,and F is a smooth random variable of the form F = f(W (h),W (e2), . . . ,W (en)), wheref ∈ C∞p (Rn). Let φ(x) denote the density function of the standard normal distribution.


Then, the expected values can be calculated by integrating with respect to x on the wholespace. Due to the orthogonality, (2.17) reduces to

E(DhF ) =∫

Rn

∂1f(x)φ(x)dx =∫

Rn

f(x)x1φ(x)dx = E(FW (h)) .

The next theorem follows as an immediate consequence of the lemma above, by apply-ing it to the product FG. It can can be regarded as an integration-by-parts formula andit will be crucial in the next parts.

Theorem 2.20 (Integration by parts). If F and G are smooth random variables andh ∈ H, then

E(GDhF ) = E(−FDhG+ FGW (h)) . (2.19)

Until now, the derivative operator was defined on S, and its value is an Ω → Hrandom variable. In the next part we will usually indicate the different spaces in thesubscripts. The Kolmogorov quotient of the set of random variables Ω → H that have abounded second moment form a Hilbert space with the following inner product, that willbe denoted by L2(Ω;H):

〈X,Y 〉L2(Ω;H) = E(〈X,Y 〉H) .

In particular, L2(Ω;H) is a Bochner space, that is, the generalization of the concept of Lp

spaces for measurable functions taking their values in an arbitrary Banach-space.The dense subspaces S0 ⊂ S and P ⊂ S is also dense in L2(Ω). We define a scalar

product and a norm on L2(Ω):

〈F,G〉1,2 = 〈F,G〉L2(Ω) + 〈DF,DG〉L2(Ω;H) ,

‖F‖21,2 = ‖F‖2L2(Ω) + ‖DF‖2L2(Ω;H) .

Define D1,2 as the closure of the domain of D in L2(Ω) with respect to the norm ‖ · ‖1,2.Then D1,2 is a Hilbert space with the scalar product 〈·, ·〉1,2. It is possible to extend D

uniquely onto D1,2. The extension is trivial; if Fn → F in L2(Ω), and DFn is convergentin L2(Ω;H), then let DF = limn→∞DFn. However, it is not trivial that the extension iswell-defined, that is, if Fn and Gn are two such series, then limn→∞DFn = limn→∞DGn.This will be guaranteed by the following theorem.

Theorem 2.21 (Closability of the operatorD). If Fn → 0 in L2(Ω) and DFn is convergentin L2(Ω;H), then limn→∞DFn = 0.

Proof. For a fixed h ∈ H, the set of random variables G ∈ S0, such that GW (h) isbounded, is dense in L2(Ω). By Theorem 2.20, we get that for such G

E( limn→∞

DhFnG) = limn→∞

E(DhFnG) = limn→∞

E(−FnDhG+ FnGW (h)) = 0 ,


because Fn → 0, and the other terms are bounded (and the dominated convergencetheorem can be used twice). Because this is valid on a dense set, we have that DhFn → 0 inL2(Ω). Since this holds for all directions h ∈ H, we obtain that DFn → 0 in L2(Ω;H).

As a consequence of the theorem above, the operator D can be extended uniquely onD1,2. We mention the following result without a proof that characterises the domain andthe effect of the operator in terms of the Wiener chaos expansion.

Theorem 2.22. Let F be a square integrable random variable with the Wiener chaosexpansion F =

∑∞n=0 Jn(F ). Then F ∈ D1,2 if and only if

E(‖DF‖2H) =∞∑n=1

n‖Jn(F )‖2L2(Ω) <∞ . (2.20)

Moreover, if (2.20) holds, then for all n ≥ 1 we have D(Jn(F )) = Jn−1(DF ).

The next result is the so-called chain rule, which can be proved by approximating therandom variable F with smooth random variables.

Theorem 2.23 (Chain rule). Let ϕ : Rm → R be a continuously differentiable functionwith bounded partial derivatives, and fix p ≥ 1. Suppose that F = (F1, . . . , Fm) is a randomvector with components belonging to D1,2. Then ϕ(F ) ∈ D1,2, and

Dϕ(F ) =m∑i=1

∂iϕ(F )DFi .

If F is a smooth random variable and we apply the derivative operator again on DF ,then we got random variable with values in H ⊗ H. By induction, DkF is a randomvariable with values in H⊗k. Similarly to ‖ · ‖1,2 we use the following notation for anyk ≥ 1:

‖F‖k,2 = ‖F‖2L2(Ω) +k∑i=1

‖DiF‖2L2(Ω;H⊗i) .

For k 6= 1 this is not a norm, just a seminorm. Nevertheless, it induces a topology onS, and the operator Dk is closable onto L2(Ω;H⊗k) analogously to the case k = 1. Thecompletion of the domain of Dk will be denoted by Dk,2.

2.4.2 The derivative in the white noise case

Assume the one dimensional white noise setting. As mentioned earlier, in this case H =L2([0, T ]) and W (t) = W (1[0,t]) is a measurable [0, T ]×Ω→ R function. The sample spacecan be regarded as Ω = C0([0, T ]) and then each ω ∈ Ω is the trajectory of the Wienerprocess W (t) = W (t, ω) = ω(t). If F (ω) = f(W (t1), . . . ,W (tn)) = f(ω(t1), . . . , ω(tn)) is asmooth random variable, then by (2.18) for any function h ∈ H, the directional derivative


is

DhF = 〈DF, h〉 =n∑i=1

∂if(W (t1), . . . ,W (tn))〈1[0,ti], h〉

= limε→0

1ε

[f(W (t1) + ε〈1[0,t1], h〉, . . . ,W (tn) + ε〈1[0,tn], h〉)− f(W (t1), . . . ,W (tn))]

= limε→0

1ε

[f(W (t1) + ε

∫ t1

0h(s)ds, . . . ,W (tn) + ε

∫ tn

0h(s)ds)− f(W (h1), . . . ,W (hn))]

= limε→0

1ε

[f(ω(t1) + ε

∫ t1

0h(s)ds, . . . , ω(tn) + ε

∫ tn

0h(s)ds)− f(ω(h1), . . . , ω(hn))]

=d

dεF (ω + ε

∫ ·0h(s)ds)|ε=0 =

d

dεF (ω + εγ)|ε=0 . (2.21)

This shows that DhF is basically a Gateaux- (or directional) derivative of F : Ω→ R fromthe direction γ(t) =

∫ t0 h(s)ds, where h ∈ L2([0, T ]). Such directions are called Cameron-

Martin directions, and the subspace of C0([0, T ]) that consists of the Cameron-Martindirections is called the Cameron-Martin space.

As mentioned above, the derivative of a random variable F ∈ D1,2 is a random variableDF ∈ L2(Ω;H), where in the white noise case H = L2([0, T ],B, µ). Thus, for each ω ∈ Ω,DF (ω) is a function in L2([0, T ],B, µ). This means that DF is a stochastic process, anelement of the canonical space L2([0, T ]×Ω,B×F , µ×P ). We can write DtF, t ∈ [0, T ]for the derivative process at time t.

We have already considered the effect of derivation on the Wiener chaos in general. Inthis particular case it is given by the next theorem.

Theorem 2.24. Let F ∈ D1,2 be a square integrable random variable having the Wienerchaos expansion F =

∑∞n=1 In(fn), where the kernels fn are symmetric functions from

L2([0, T ]m). Then, the derivative process can be computed as

DtF =∞∑n=1

nIn−1(fn(·, t)) . (2.22)

Proof. Let us assume first that the expansion of F consists only one term: F = Im(fm),such that fm is a symmetric and elementary function of the form (2.7). Then the derivativeof F at time t is

DtF = DtIm(fm) = Dt

m∑i1,...,im=1

ai1···imW (Ai1) · · ·W (Aim)

=

m∑j=1

m∑i1,...,im=1

ai1···imW (Ai1) · · ·1Aij(t) · · ·W (Aim) =

= mm∑

i1,...,im=1

ai1···imW (Ai1) · · ·1Aij(t) · · ·W (Aim)︸︷︷︸

Im−1(fm(·,t,·))

=

= mIm−1(fm(·, t)) .


The general case follows from the usual density and linearity properties.

Let FA denote the σ-field generated by the random variables W (B), B ⊂ A,B ∈B, µ(B) < ∞. The derivative of the conditional expected value of a random variablewith respect to FA can be calculated using the next theorem.

Theorem 2.25. Let F ∈ D1,2 and A ∈ B. Then the conditional expectation E(F |FA) alsobelongs to the space D1,2, and we have

Dt(E(F |FA)) = E(DtF |FA)1A(t) .

2.5 The divergence operator and the Skorohod integral

In this section the divergence operator will be introduced in the frame work of a Gaussianisonormal process over a Hilbert space H. It is assumed that W is defined on completeprobability space (Ω,F , P ), where F is generated by W . Remember that the derivativeoperator D is a closed and unbounded operator with values in L2(Ω;H) defined on thedense subset D1,2 of L2(Ω).

Definition 2.26. The divergence operator δ is defined as the adjoint of operator D. Thismeans that δ is an unbounded operator on L2(Ω;H) with values in L2(Ω) such that

(i) The domain of δ, denoted by Domδ, is the set of H-valued square integrable randomvariables u ∈ L2(Ω;H), such that

|〈DF, u〉L2(Ω;H)| = |E(〈DF, u〉H)| ≤ c‖F‖L2(Ω) , (2.23)

for all F ∈ D1,2, where c is some constant depending on u.

(ii) If u ∈ Domδ, then δ(u) is the element of L2(Ω) such that

〈F, δ(u)〉L2(Ω) = E(Fδ(u)) = E(〈DF, u〉H) = 〈DF, u〉L2(Ω;H) (2.24)

Remark. δ is a linear, closed, unbounded and densely defined operator. The existence ofthis adjoint follows from the Riesz representation theorem for linear operators on Hilbertspaces. From (2.24) it follows that E(δ(u)) = 0.

Let us denote by SH ⊂ L2(Ω;H) the set of H valued smooth random variables of theform

F =n∑j=1

Fjhj ,

where Fj ∈ S and hj ∈ H for all 1 ≤ j ≤ n. The derivative of a random variable F ∈ SHis a square integrable random variable with values in the Hilbert space H⊗H. We definethe following seminorm on SH:

‖F‖21,2 = ‖F‖2L2(Ω;H) + ‖DF‖2L2(Ω;H⊗H) .

The completion of S with respect to this seminorm is denoted as D1,2(H).We have the following results on the divergence operator (proofs are in [9]).


Theorem 2.27. The space D1,2(H) is included in Domδ. If u, v ∈ D1,2(H), then

〈δ(u), δ(v)〉L2(Ω) = 〈u, v〉L2(Ω;H) + 〈Du,Dv〉L2(Ω;H⊗H) .

Theorem 2.28. Suppose u ∈ D1,2(H), and Dhu ∈ Domδ for some h ∈ H. Then Dh(δ(u))exists, and

Dh(δ(u)) = 〈u, h〉H + δ(Dhu) . (2.25)

Remark. The equation (2.25) is basically a Heisenberg commutativity relationship. It canbe written in the following form using commutator brackets:

[Dh, δ]u = 〈u, h〉H .

Theorem 2.29 (Integration by parts). Let F ∈ D1,2, u ∈ Domδ, such that Fu ∈ L2(Ω;H).Then Fu ∈ Domδ, and

δ(Fu) = Fδ(u)− 〈DF, u〉H

2.5.1 The Skorohod integral

Let us consider the divergence operator in the white noise setting. As mentioned above,in this case L2(Ω;H) ∼= L2([0, T ]×Ω). Similarly to L2(Ω), this Hilbert space has a Wienerchaos expansion. In this case each u(t) ∈ L2([0, T ]×Ω) can be decomposed in the followinginfinite sum:

u(t) =∞∑n=0

In(fn(·, t)) , (2.26)

where for each n ≥ 1, fn ∈ L2([0, T ]n+1) is a symmetric function in the first n variables.Moreover, we have the following equality

‖u‖2L2([0,T ]×Ω) = E

(∫ T

0u(t)2dµ(t)

)=∞∑n=0

n!‖fn‖2L2([0,T ]n+1) .

Theorem 2.30. Let u ∈ L2([0, T ]× Ω) with the expansion (2.26). If u ∈ Domδ then

δ(u) =∞∑n=0

In+1(fn) . (2.27)

Conversely, if (2.27) converges in L2(Ω), then u ∈ Domδ.

Remark. This is called as the Skorohod integral of u with respect to W (t). In many caseswe will denote this kind of integral as

∫ T0 u(t)δW (t).

Proof. Let G = In(g) be a multiple stochastic integral of an arbitrary symmetric function


g. Then

〈u,DtG〉L2([0,T ]×Ω) = E

(∫ T

0u(t)DtGµ(dt)

)=∞∑m=0

∫ T

0E(Im(fm(·, t))nIn−1(g(·, t)))dµ(t)

=∫ T

0E(In−1(fn−1(·, t))nIn−1(g(·, t)))dµ(t)

= n(n− 1)!∫ T

0〈fn−1(·, t), g(·, t)〉dµ(t)

= n!〈fn−1, g〉L2([0,T ]n) = n!〈fn−1, g〉L2([0,T ]n)

= E(In(fn−1)In(g)) = E(In(fn−1)G) .

If u ∈ Domδ, then from the adjoint property (2.24) we have that

〈δ(u), G〉L2(Ω) = E(δ(u)G) = E(In(fn−1)G)

for every multiple stochastic integral G = In(g). From this it follows that In(fn−1) andδ(u) coincide on the n-th Wiener chaos, and hence In(fn−1) is the projection of δ(u).Therefore, (2.26) converges in L2(Ω) and since the direct sum of the Wiener chaoses isL2(Ω), the sum is just δ(u).

Conversely, suppose that the series converges to V in L2(Ω). Then, from the compu-tations above it follows that

E

(∫ T

0u(t)Dt

(N∑n=0

In(gn)

)µ(dt)

)= E(V

N∑n=0

In(gn))

for all N ≥ 0. From this and the Cauchy-Schwartz inequality it follows that

|〈u(t), DtF 〉L2([0,T ]×Ω)| = |E(∫ T

0u(t)DtFµ(dt))| ≤ ‖V ‖2‖F‖2

for any random variable F , that has a finite Wiener chaos expansion. Such variables aredense in L2(Ω), and therefore, this inequality holds for any random variable F ∈ D1,2.This means that by Definition 2.26, u belongs to Domδ.

Next, it will be shown that the Skorohod integral can be regarded as an extensionof the Ito integral to integrands that are not necessarily Ft-adapted, where Ft is thestandard filtration of the Brownian motion. In particular, the next result states that thetwo integrals coincide when u(t) is adapted.

Theorem 2.31. Let u(t) ∈ L2([0, T ]×Ω), and suppose u(t) is Ft-adapted. Then u ∈ Domδand ∫ T

0u(t)δW (t) =

∫ T

0u(t)dW (t) .


Lemma 2.32. Let u(t) ∈ L2([0, T ] × Ω) be a stochastic process with the Wiener chaosexpansion 2.26. Then u(t) is Ft-adapted if and only if

fn(t1, . . . , tn, t) = 0 if t < maxiti .

Proof of Theorem 2.31. By Lemma 2.32 and taking into account that fn is symmetricalin the first n variables, we have that its symmetrization is

fn(t1, . . . , tn+1) =1

n+ 1

n+1∑i=1

fn(. . . , ti−1, ti+1, . . . , ti) =1

n+ 1fn(. . . , tj−1, tj+1, . . . , tj) ,

(2.28)where tj = max1≤i≤n+1 ti. Using again Lemma 2.32, we have the following expression forthe norm of fn:

‖f‖2L2([0,T ]n+1) = (n+ 1)!∫Sn+1

f2n(x1, . . . , xn+1)dx1 · · · dxn+1

=(n+ 1)!(n+ 1)2

∫Sn+1

f2n(x1, . . . , xn+1)dx1 · · · dxn+1

=n!

n+ 1

∫ T

0

(∫ t

0

∫ xn

0· · ·∫ x2

0f2n(x1, . . . , xn, t)dx1 · · · dxndt

)=

n!n+ 1

∫ T

0

(∫ T

0

∫ xn

0· · ·∫ x2

0f2n(x1, . . . , xn, t)dx1 · · · dxndt

)=

1n+ 1

∫ T

0‖fn(·, t)‖2L2([0,T ]n)dt .

Hence∞∑n=0

(n+ 1)!‖fn‖2L2([0,T ]n+1) =∞∑n=0

n!∫ T

0‖fn(·, t)‖L2([0,T ]n)dt

=∫ T

0

∞∑n=0

n!‖fn(·, t)‖L2([0,T ]n)dt = E(∫ T

0u2(t)dt) = ‖u‖2L2([0,T ]×Ω) <∞ .

Therefore u ∈ Domδ. Finally, we again apply (2.28) to prove the equivalence of theintegrals:∫ T

0u(t)dW (t) =

∞∑n=0

∫ T

0In(fn(·, t))dW (t)

=∞∑n=0

∫ T

0

(n!∫

0≤t1≤···≤tn≤tfn(t1, . . . , tn, t)dW (t1) · · · dW (tn)

)dW (t)

=∞∑n=0

∫ n

0!(n+ 1)

∫0≤t1≤···≤tn≤tn+1

fn(t1, . . . , tn, tn+1)dW (t1) · · · dW (tn)dW (tn+1)

=∞∑n=0

(n+ 1)!1

(n+ 1)!In+1(fn) =

∫ T

0u(t)δW (t) = δ(u) .


2.6 The Clark-Ocone formula

The following result provides an integral representation of any square integrable functionalof the Brownian motion. Set FT = σW (s), s ∈ [0, T ].

Theorem 2.33 (Ito representation theorem). Let F ∈ L2(Ω,FT , P ). Then there exists aunique process u ∈ L2

a([0, T ]× Ω) such that

F = E(F ) +∫ T

0u(t)dW (t) .

Proof. To prove the theorem it is enough to show that any zero-mean square inte-grable random variable G that is orthogonal to all stochastic integrals

∫ T0 u(t)dW (t),

u ∈ L2a([0, T ] × Ω) must be zero. Since L2

a([0, T ] × Ω) is complete, this implies that thenG = 0 almost everywhere. Take Mu(t) = exp(

∫ t0 u(s)dW (s)− 1

2

∫ t0 u

2(s)ds). Applying theIto formula we deduce that

Mu(t) = 1 +∫ t

0Mu(s)u(s)dW (s) .

Therefore, the random variable G is orthogonal to the exponentials

E(h) = exp(∫ T

0h(s)dW (s)− 1

2h2(s)ds) ,

where h ∈ L2([0, T ]). By Lemma 2.8, such random variables form a complete subset ofL2(Ω,FT ) and we conclude that G is zero in L2(Ω,FT ).

The next theorem is very important, since it gives the process u explicitly in the caseswhen F ∈ D1,2.

Theorem 2.34 (Clark-Ocone formula). Let F ∈ D1,2 be FT measurable. Then

F = E(F ) +∫ T

0E(DtF |Ft)dW (t) .

Proof. Write F =∑∞

n=0 In(fn) with fn ∈ L2([0, T ]n). Applying Proposition 2.16 andTheorem 2.24 we deduce∫ T

0E(DtF |Ft)dW (t) =

∫ T

0E(∞∑n=1

In−1(fn(·, t))|Ft)dW (t)

=∫ T

0

∞∑n=1

nIn−1(fn(t1, . . . , tn−1, t)1[0,t]n−1(t1, . . . , tn−1))dW (t)

=∞∑n=1

n

∫ T

0In−1(fn(t1, . . . , tn−1, t)1[0,t]n−1(t1, . . . , tn−1))dW (t)

=∞∑n=1

n1nIn(fn) =

∞∑n=0

In(fn)− I0(f0) = F − E(F ) ,

where at fourth equality we used Theorem 2.27 and the fact that the Skorohod and theIto integrals coincide in the case of adapted processes, and here this is the case because ofthe indicator function.


Suppose that

W (t) = W (t) +∫ t

0θ(s)ds ,

where θ is an adapted and measurable process such that∫ T

0 θ(t)2dt < ∞ almost surely.By Girsanov theorem, the process W is a Brownian motion under the new probabilitymeasure Q defined on FT by

dQ(ω) = ZT (ω)dP (ω) , (2.29)

where

Zt = exp(−∫ t

0θ(s)dW (s)− 1

2

∫ t

0θ(s)2dt

). (2.30)

Let denote by EQ the expectation with respect to Q. Then a generalization of the Clark-Ocone formula is valid.

Theorem 2.35 (Generalized Clark-Ocone formula). Let F ∈ D1,2 be an FT -measurablerandom variable such that

(i) EQ(F 2) + EQ(∫ T

0 (DtF )2dt) <∞,

(ii) EQ

(F 2∫ T

0

(θ(t) +

∫ Tt Dtθ(s)dW (s) +

∫ Tt θ(s)Dtθ(s)ds

)2dt

)<∞.

Then

F = EQ(F ) +∫ T

0EQ

(DtF − F

∫ T

tDtθ(s)dW (s)|Ft

)dW (t) .

It is important to note that it is not possible to obtain a representation of F as anintegral with respect to W (t) simply by applying the Clark-Ocone formula, because Fis only assumed to be FT -measurable, not FT -measurable, where FT is the filtrationgenerated by W (t). In general FT ⊆ FT , but usually FT 6= FT . Nevertheless, the integralwith respect to W (t) makes sense, because W (t) is a martingale and with more advancedtechniques it is possible to prove the generalized Clark-Ocone formula for this general case.

Chapter 3

Applications

In this chapter three important applications of Malliavin calculus are briefly surveyed.First, the probabilistic version of Hormander’s condition for regularity is summarized.Then, bounds on the Kolmogorov distance of two random variables based on the Malliavinderivatives are given. Finally, the application of the generalized Clarck-Ocone formula infinancial mathematics is discussed.

3.1 Stochastic differential equations

One of the most important applications of Malliavin calculus is in the theory of stochasticdifferential equations. In this section we briefly summarize some of the key results of thisarea. Conditions for the smoothness of random vectors and the solutions of stochasticdifferential equation are investigated.

3.1.1 Absolute continuity of distributions

Given a random vector F = (F1, . . . , Fm) that is measurable with respect to an underlyingGaussian process W (h), h ∈ H, one may be interested in the regularity of its distribu-tion. Namely, conditions for the absolute continuity of (multivariate) distributions withrespect to the Lebesgue measure can be investigated.

Definition 3.1. The Borel measure µ (that is, here, the distribution or law of a randomvariable) is absolute continuous with respect to the Lebesgue measure given the followingequivalent conditions:

1. µ(A) = 0 for every A ∈ B, for which λ(A) = 0,

2. for every ε > 0, there is a δ > 0, such that µ(A) < ε for all A ∈ B, λ(A) < δ,

3. there is a Lebesgue integrable function g, such that µ(A) =∫A gdλ for all A ∈ B.

Remark. From now on we will call such measures absolute continuous or just continuous.

22

CHAPTER 3. APPLICATIONS 23

Proposition 3.2. Let µ be a finite measure on Rm. Suppose that there are constants cifor all 1 ≤ i ≤ m, such that for all ϕ ∈ C∞(Rm) the following inequality holds:∣∣∣∣∫

Rm

∂iϕdµ

∣∣∣∣ = ci‖ϕ‖∞ . (3.1)

Then µ is absolute continuous.

Assume, that all the components Fi are in D1,2, and take ϕ : Rm → R as above. Thenwe can investigate the conditions of this proposition for ϕ(F ).

Definition 3.3. The Malliavin (covariance) matrix of a random vector F is the matrix

ΛF = (〈DFi, DFj〉H)1≤i,j≤m .

By the chain rule for the derivative operator we have that

〈Dϕ(F ), DFj〉H = 〈m∑i=1

∂iϕ(F )DFi, DFj〉H

=m∑i=1

∂iϕ(F )〈DFi, DFj〉H = (∇ϕ(F )ΛF )j .

Assume Λ−1F exists almost surely. Then

∇ϕ(F ) = (〈Dϕ(F ), DF1〉H), . . . , 〈Dϕ(F ), DFm〉H)Λ−1F .

From this the partial derivatives of ϕ(F ) can be expressed as

∂iϕ(F ) =m∑j=1

〈Dϕ(F ), (Λ−1F )jiDFj〉H .

Applying the definition of δ and estimating the, we have the following for the expectedvalue of the partial derivative

E(∂iϕ(F )) =m∑j=1

E(〈Dϕ(F ), (Λ−1F )jiDFj〉H)

=m∑j=1

E(ϕ(F )δ((Λ−1F )jiDFj))

≤ ‖ϕ‖∞m∑j=1

E(δ((Λ−1F )jiDFj)) ,

given (Λ−1F )jiDFj ∈ Domδ. If, in addition, E(δ((Λ−1

F )jiDFj)) is finite for all 1 ≤ i, j ≤ m,then by Proposition 3.2 the distribution of F is absolute continuous. It can be provedthat this is always the case, and this provides the following very important theorem.

Theorem 3.4. Let F = (F1, . . . , Fm) be a random vector such that

(i) Fi ∈ D1,2 for all 1 ≤ i ≤ m.


(ii) The matrix ΛF is invertible a.s.

Then the distribution of F is absolute continuous on Rm.

This is a very deep theorem, and we omit the proof because it uses a lot of advancedtechniques. Lastly, we mention the following result, that even more is true when strongerconditions hold for F and ΛF .

Theorem 3.5. Let F = (F1, . . . , Fm) be a random vector such that

(i) Fi ∈ D∞,2 for all 1 ≤ i ≤ m.

(ii) (DetΛF )−1 ∈ ∩p≥1Lp(Ω).

Then F has a C∞-density.

3.1.2 Lie bracket and Hormander’s condition

Assume that V and W are smooth vector fields on Rm. This means that V =∑m

i=1 Vi(x)ei

and W =∑m

i=1Wi(x)ei, where e1, . . . , em is an orthonormal basis of Rm. There is a useful

notation due to Einstein for such sums that we will use use thereafter. According to thisconvention, when an index variable appears twice in a single term, once in an upper(superscript) and once in a lower (subscript) position, it implies that we are summing overall of its possible values. In particular, we can omit the sum sign above and write justV = V i(x)ei and W = W i(x)ei. The covariant derivative of V in the direction of W isdefined as

∇WV =m∑i=1

m∑j=1

W j∂jViei = W j∂jV

iei .

It is easy to check that ∇WV = ∂V ·W , where ∂V is the Jacobian of V.In many cases, the vector field V is identified with a first order differential operator

by thinking about it as

V =m∑i=1

V i(x)∂i = V i(x)∂i .

In this case there is an ordinary differential equation on Rm given by

dX

dt= V X ,

and this equations has (at least a local) solution for every x ∈ Rm, such that Xx(0) = x.By setting

Vt(x) = Xx(t)

we have a (local) 1-parameter group or flow. Let us define the Lie bracket of two smoothvector field V and W by

[V,W ] = ∇VW −∇WV .

It can be shown that the Lie bracket measures how much the two associated flows (Vt andWt) lack to commute.


Suppose A0, . . . , Ad are smooth (C∞) vector fields and consider the associated secondorder differential operator on U ⊂ Rm

L =12

d∑j=1

(Aj)2 +A0 . (3.2)

An important problem in the theory of partial differential equations is to determine suffi-cient conditions on the vector fields Ai under which the distribution (generalized function)solutions u of

Lu = f

are always smooth on a V ⊂ U , whenever f is smooth on V . Such differential operatorsare called hypoelliptic.

It can be shown ([12] ch. 3, 15.6) that an equivalent condition to hypoellipticity isthat the Cauchy problem

∂tu(t, x) = Lu(t, x), t > 0, x ∈ U, u(t, x) = f(x) as t→ 0 .

has a smooth fundamental solution. That is a function p(t, x, y) : R+ × U × U → R, suchthat p is smooth in the space variables for each t and

u(t, x) =∫Up(t, x, y)f(y)dy

satisfies the Cauchy problem above. It is known that a smooth solution exists in the ellipticcase in which the matrix A = (Aij)1≤j≤d,1≤i≤m is such that AA∗ is invertible everywhere.

Definition 3.6. The vector fields A0, . . . , Ad satisfy Hormander’s condition at x ∈ Rm ,if the vector fields

• A1, . . . , Ad

• [Ai, Aj ], 0 ≤ i, j ≤ d

• [Ai, [Aj , Ak]], 0 ≤ i, j, k ≤ d...

span Rm at x.

Theorem 3.7 (Hormander, [4]). If the coefficients of the differential operator (3.2) satisfyHormander’s condition at each point x ∈ U , then L is hypoelliptic.

3.1.3 Absolute continuity under Hormander’s condition

Suppose that (Ω,F , P ) is the canonical probability space of the d-dimensional Brownianmotion W i(t), t ∈ [0, T ], 1 ≤ i ≤ d. We consider the induced isonormal Gaussian processW with the underlying Hilbert space H = L2([0, T ]; Rd). Let Aj , B : Rm → Rm, 1 ≤ j ≤ dbe measurable functions (vector fields) that have bounded derivatives of all orders. Let us


denote by X(t), t ∈ [0, T ] the solution of the following m-dimensional stochastic differentialequation:

X(t) = x0 +∫ t

0B(X(s))ds+

d∑j=1

∫ t

0Aj(X(s))dW j(s) (3.3)

Now consider the coefficients of the stochastic differential equation above. Set

A0 =[Bi(x)− 1

2Ajl (x)∂jAil(x)

]ei = B − 1

2

d∑l=1

∇AlAl ,

which is again a vector field. Using this it is easy to express (3.3) in terms of a Stratonovich-integral instead of an Ito integral:

X(t) = x0 +∫ t

0A0(Xs)ds+

∫ t

0Aj(Xs)∂W j(s) .

The next theorem can be regarded as the stochastic version of Hormander’s theorem,but has a significantly weaker assumption.

Theorem 3.8 (Malliavin). If the coefficients A0, . . . , Ad satisfy Hormander’s condition atthe initial value x0, then the distribution of X(t) is absolute continuous for all t ∈ (0, T ].

We will outline the main components of the proof of this theorem following [1] and [3].For this the following preliminary results are needed.

Lemma 3.9. Given A0, . . . , Ad are as above, there exists a unique solution X(t) = Xi(t)eiof (3.3) which is continuous in t, and Xi(t) ∈ D1,2 for all 1 ≤ i ≤ m and t ∈ [0, T ].

We will denote byΛji (t) = 〈DXi(t), DXj(t)〉H

the Malliavin matrix of the solution at time t. Now fix an arbitrary t ∈ (0, T ]. It can beshown that the matrix Λ = Λ(t) can be written in the following form

Λ = Z−1t

[∫ t

0ZsA(X(s))A>(X(s))Z>s ds

]Z−>t ,

where A = (A1, . . . , An) and Z is defined by the following Stratonovich differential equa-tion:

Zt = I −∫ t

0Zs∂A0(X(s))ds−

d∑i=1

∫ t

0Zs∂Ai(X(s))∂W i(s) .

Here I is the m×m identity matrix and ∂Ai(X(s)) is the Jacobian of Ai at X(s). (Notethat the multiplication is matrix multiplication!)

Lemma 3.10. Suppose that V is a smooth vector field on Rm and τ is a stopping timesuch that

ZsV (X(s)) = 0, ∀s ∈ [0, τ ] . (3.4)

Then for each 0 ≤ j ≤ d

Zs[Aj , V ](X(s)) = 0, ∀s ∈ [0, τ ] .


Proof. From (3.4) it follows that

d(ZsV (X(s))) = 0, ∀s ∈ [0, τ ] .

Then, using Ito formula for the left-hand side and taking into account that ∂Ai · V − ∂V ·Ai = ∇AiV −∇VAi = [V,Ai] we have that

Zs[V,A0](X(s))ds+d∑j=1

Zs[V,Aj ](X(s))∂W j(s) = 0, ∀s ∈ [0, τ ] .

This Stratonovich stochastic differential equation can be rewritten as an Ito differentialequation as Zs[V,A0](X(s)) +

12

d∑i,j=1

Zs[[V,Ai], Aj ](X(s))

ds+∑i=1

Zs[V,Ai](X(s))dW i(s) = 0, ∀s ∈ [0, τ ] .

From this it follows that

Zs[V,Aj ](X(s)) = 0, ∀s ∈ [0, τ ], 1 ≤ i ≤ d ,

and

Zs[V,A0](X(s)) +12

d∑i,j=1

Zs[[V,Ai], Aj ](X(s)) = 0, ∀s ∈ [0, τ ] .

With essentially the same argument as the first it can be shown that Zs[[V,Ai], Aj ](X(s)) =0 on [0, τ ] for all 1,≤ i, j ≤ d. Using this in the second relation gives

Zs[V,A0](X(s)) = 0, ∀s ∈ [0, τ ] .

Remark. The assumption and the conclusion of Lemma 3.10 could have replaced by thestatements 〈ZsV (X(s)), y〉 = 0 and 〈Zs[Aj , V ](X(s)), y〉 = 0 on[0, τ ] for any y ∈ Rm and0 ≤ j ≤ d.

Let us denote by RanΛ(t) ⊂ Rm the image of the random and time-dependent matrixΛ(t).

Lemma 3.11.

SpanA1(x0), . . . , Ad(x0), [Ai, Aj ](x0), [Ai, [Aj , Ak]](x0), . . . ,

0 ≤ i, j ≤ d, 0 ≤ i, j, k ≤ d, . . . ⊆ RanΛ(t)

for and t ∈ (0, T ] almost surely.


Proof. For each 0 < s < t define

R(s) = SpanZuAi(X(s)), 0 ≤ u ≤ s, 1 ≤ i ≤ d

andR = R(ω) = ∩s>0R(s) .

It can be shown that R(t) = RanΛ(t). Therefore, by the Blumenthal 0 − 1 law thereexists a (deterministic) set R such that R(ω) = R a.s. Suppose that y ∈ R⊥. Then withprobability 1 there exists τ > 0 such that R(s) = R for s ∈ [0, τ ]. This means that

〈ZsAi(X(s)), y〉 = 0, ∀s ∈ [0, τ ], 1 ≤ i ≤ d ,

or simply y⊥ZsAi(X(s)). Moreover, by iterating Lemma 3.10 we get

y⊥Zs[Aj , Ak](X(s)), y⊥Zs[[Aj , Ak], Al](X(s)), . . . , ∀s ∈ [0, τ ] .

Evaluating this at s = 0 shows that y ∈ (SpanH)⊥, where H is the set of vectors listed inHormander’s condition. Therefore H ⊆ R ⊆ R(t) = RanΛ(t).

Corollary 3.12. Λ(t) is invertible with probability 1.

Combining Theorem 3.4 with this result Theorem 3.8 follows.

3.2 Relation to Stein’s method

In this section we investigate to connection of Malliavin calculus to Stein’s method for thenormal approximation of probability distributions. This part is based heavily on [8].

3.2.1 Stein’s lemma

Let N ∼ N(0, 1) be a standard Gaussian random variable. It is well known that themoments of N are

µp = E(Np) =

(p− 1)!! = p!

2p/2(p/2)!if p is even

0 if p is odd

It can be shown that the sequence µp : p ≥ 1 is completely determined by the recurrencerelation

µ1 = 0, µ2 = 1 and µp = (p− 1) · µp−2 for every p ≥ 3 . (3.5)

Now introduce the notation fp(x) = xp. Then the relation (3.5) can be restated as

E(N · fp−1(N)) = E(f ′p−1(N)), ∀p ≥ 1 . (3.6)

By using a standard argument based on polynomial approximations, one can easily provethat (3.6) continues to hold if one replaces fp with a sufficiently smooth function f . Con-versely, if for a random variable Z (3.6) holds, then it has moments E(Zp) = µp for allp ≥ 1. Let us recall that the distribution of a N(0, 1) random variable is uniquely deter-mined by its moments. Combining these facts with some additional arguments yields tothe following characterization of the normal distribution.


Theorem 3.13 (Stein’s lemma). Let Z be a generic random variable. Then Z ∼ N(0, 1)if and only if

E(Zf(Z)− f ′(Z)) = 0

for every continuous and piecewise differentiable function f satisfying the relationE|f ′(N)| <∞.

If two random variables does not follow exactly the same distributions, it is interestingto investigate how “far” are they from each other in some sense.

Definition 3.14. The Kolmogorov distance of two real-valued random variables Y and Zis defined by

dK(Y,Z) = supz∈R|P (Y ≤ z)− P (Z ≤ z)| .

Stein has given the following upper bound on the Kolmogorov distance, which is closelyrelated to Theorem 3.13 and can be proved using real analytic techniques.

Theorem 3.15 (Stein’s bound). Let N ∼ N(0, 1) and Z be an arbitrary random variable.Then

dK(Z,N) ≤ supf|E(Zf(Z)− f ′(Z))| , (3.7)

where the supremum is taken over the class of all Lipschitz functions that are bounded by√2π4 and whose Lipschitz constant is less or equal to 1.

3.2.2 Ornstein-Uhlenbeck operators

Recall that Jn : L2(Ω) → Hn is the orthogonal projection onto the n-th Wiener chaos.Let F ∈ L2(Ω) be a random variable. We define the operator L as follows:

LF =∞∑n=0

−nJnF .

It is easy to see that the domain of L is

DomL = F ∈ L2(Ω) :∞∑n=1

n2‖Jn(F )‖2L2(Ω) <∞ = D2,2 .

The operator L is called the generator of the Ornstein-Uhlenbeck semigroup. It is impor-tant to mention that L is an unbounded symmetric operator. That is, 〈F,LG〉L2(Ω) =E(FLG) = E(GLF ) = 〈LF,G〉L2(Ω) for all F,G ∈ DomL. There is an important relationbetween the operators D, δ and L, which is stated in the next result.

Proposition 3.16. A random variable F belongs to D2,2 if and only if F ∈ Dom(δD)(i.e. F ∈ D1,2 and DF ∈ Domδ) and, in this case,

δDF = −LF .


For any F ∈ L2(Ω) let us define another operator L−1 as follows:

L−1F =∞∑n=1

− 1nJnF .

It is easy to see from the definition that L−1F ∈ DomL for any F ∈ L2(Ω). Moreover,LL−1F = F − E(F ). Therefore, L−1 is called the pseudo-inverse of L.

Using the mentioned properties of L and L−1 it is possible to continue the idea ofStein’s bound. Let f : R→ R be a C1 function with bounded derivative and F,Z ∈ D1,2,such that E(Z) = 0. Then, using the definition of δ and the properties of D one has

E(Zf(F )) =E(LL−1Z · f(F )) = E(δD(−L−1Z) · f(Z))

=E(〈Df(F ),−DL−1Z〉H)

=E(f ′(F )〈DF,−DL−1Z〉H) . (3.8)

3.2.3 Bounds on the Kolmogorov distance

Suppose Z is a centered and Malliavin-differentiable random variable. Then, a bound onits Kolmogorov distance from the normal distribution can be given using the results above.

Theorem 3.17. Let Z ∈ D1,2 be such that E(Z) = 0 and Var(Z) = 1. Then, forN ∼ N(0, 1)

dK(Z,N) ≤√

Var(〈DZ,−DL−1Z〉H)

Proof. In view of Theorem 3.15 it is enough to show, that for every Lipschitz function f

with Lipschitz constant less than or equal to 1,

|E(Zf(Z)− f ′(Z))| ≤√

Var(〈DZ,−DL−1Z〉H) .

First consider f : R→ R such that f ∈ C1 and ‖f ′‖∞ ≤ 1. Then (3.8) yields

E(Zf(Z)) = E(f ′(Z)〈DZ,−DL−1Z〉H) ,

so that

|E(f ′(Z))− E(Zf(Z))| = |E(f ′(Z)(1− 〈DZ,−DL−1Z〉H))| ≤ E|1− 〈DZ,−DL−1Z〉H| .

By a standard approximation argument, one sees that the inequality

|E(f ′(Z))− E(Zf(Z))| ≤ E|1− 〈DZ,−DL−1Z〉H|

continues to hold when f is Lipschitz with constant less than or equal to 1. Hence, bycombining the previous estimates with Theorem 3.15, we infer that

dK(Z,N) ≤ E|1− 〈DZ,−DL−1Z〉H| ≤√E(1− 〈DZ,−DL−1Z〉H)2 .


Finally, by applying (3.8) to Z and the function f(z) = z we get

E(〈DZ,−DL−1Z〉H) = E(Z2) = 1 .

Therefore,E(1− 〈DZ,−DL−1Z〉H)2 = Var(〈DZ,−DL−1Z〉H) .

Lemma 3.18. Suppose Z belongs to the n-th Wiener chaos Hn. Then

Var(1n‖DZ‖2H) ≤ n− 1

3n(E(Z4)− 3) .

Theorem 3.19. Let Z belong to the n-th Wiener chaos for some n ≥ 2. Suppose moreoverthat Var(Z) = 1. Then

dK(Z,N) ≤√n− 1

3n(E(Z4)− 3) .

Proof. Since L−1Z = − 1nZ we have 〈DZ,−DL−1Z〉H = 1

n‖DZ‖2H. Then, applying Theo-

rem 3.17 and Lemma 3.18 the result follows.

Using this result it is not so difficult to prove the last theorem of this section that hasbeen used in many approximation problems of statistics and probability theory.

Theorem 3.20. Let Zk be a sequence of random variables belonging to the n-th Wienerchaos for some n ≥ 2. Assume that Var(Zk) = 1 for all k. Then, as k →∞, the followingthree assertions are equivalent:

(i) Zk → N ∼ N(0, 1) in distribution;

(ii) E(Z4k)→ E(N4) = 3;

(iii) Var( 1n‖DZk‖

2H)→ 0.

3.3 Financial mathematics

Finally, some interesting applications in financial mathematics will be discussed briefly.This section is based on the lecture notes of Brent Øksendal [10].

3.3.1 Investments

Let us assume that on a market two types of investments exist:

1. A safe investment (e.g. a bond) with price dynamics

dA(t) = ρ(t)A(t)dt , (3.9)


2. A risky investment (e.g. a stock) with price dynamics

dS(t) = µ(t)S(t)dt+ σ(t)S(t)dW (t) , (3.10)

where ρ(t, ω) is the interest rate, µ(t, ω) is the drift and σ(t, ω) is the volatility and allof them are Ft-adapted stochastic processes. In the following we will not specify furtherconditions, but simply assume that these processes are sufficiently nice. Moreover, inmany cases we will not indicate the dependence on ω.

If we denote by ξ(t) and η(t) the capital invested at time t in the safe and riskyinvestments respectively, then the value of this portfolio is given by

V (t) = ξ(t)A(t) + η(t)S(t) . (3.11)

The portfolio is called self-financing if

dV (t) = ξ(t)dA(t) + η(t)dS(t) .

This means that change in the value of the portfolio comes only from the change in pricesof the assets, that is, no funds are borrowed or withdrawn from the portfolio at any time.In this case, by expressing ξ(t) from (3.11) and using (3.9) we get that

dV (t) = ρ(t)(V (t)− η(t)S(t))dt+ η(t)dS(t) .

Then, by (3.10) this can be written as

dV (t) = (ρ(t)V (t) + (µ(t)− ρ(t))η(t)S(t))dt+ σ(t)η(t)S(t)dW (t) . (3.12)

An important concept in financial mathematics is the arbitrage, which can be consid-ered as a strategy that allows to make profit out of nothing without taking any risk. Givena general model one may ask the following questions.

1. If we have a model for evolution of prices, how can we tell if there are arbitrageopportunities?

2. If we know that there are no arbitrage opportunities in the market, how do we pricea claim, such as an option?

From these questions we will investigate the second one.Suppose that we are required to find a portfolio (ξ(t), η(t)) which leads to a given value

(claim)V (T, ω) = F (ω) (3.13)

at a given deterministic future time T , where the given F (ω) is FT -measurable. A generalquestion is that what is the initial fortune V (0) that is needed to achieve this and whatportfolio (ξ(t), η(t)) should be used? This problem arises in the pricing of claims (like calland put options). If such portfolio exists, which is called a replicating portfolio, and itis satisfies some further properties (admissibility), then the claim F is attainable. If any


claim is attainable, then the market model is called complete. If a model is complete, thenany claim can be priced by no-arbitrage considerations.

Since (V (t), η(t)) describes the portfolio, it is possible to consider this pair as theunknown Ft adapted process. Then (3.12) and (3.13) form together a stochastic backwarddifferential equation (SBDE) problem, with the given final value F (T, ω). The solution ofthis is the pair (V (t), η(t)) for 0 ≤ t ≤ T . Since V (t) is Ft adapted, it follows that V (0)is F0 adapted and therefore it is a constant. The general theory of SBDE gives that thisproblem has a unique solution but it does not give this solution explicitly. It is importantto mention that generally this solution is not guaranteed to be admissible. Next, we willuse the generalized Clark-Ocone formula to obtain the solution.

The main difficulty in giving the replicating portfolio is the drift µ(t) of the riskyinvestment. This process, in general, can fluctuate quite fast and it is hard to predict it.On the other hand, on a usual market the interest rate ρ(t) behaves much nicer, and variesslower with the time. Our aim is to eliminate µ(t) from the equations and substitute itwith ρ(t).

Defineθ(t) =

µ(t)− ρ(t)σ(t)

and

W (t) = W (t) +∫ t

0θ(s)ds .

Then, as mentioned in Chapter 2, W (t) is a Brownian motion with respect to the measureQ defined by (2.29)-(2.30). Rewriting (3.12) as an SBDE driven by W (t) one gets

dV (t) = (ρ(t)V (t) + (µ(t)− ρ(t))η(t)S(t))dt+ σ(t)η(t)S(t)dW (t)

− σ(t)η(t)S(t)σ−1(t)(µ(t)− ρ(t))dt ,

which reduces todV (t) = ρ(t)V (t)dt+ σ(t)η(t)S(t)dW (t) . (3.14)

In addition, let us define for the moment

U(t) = e−R t0 ρ(s)dsV (t) .

Then, (3.14) with respect to U(t) becomes

dU(t) = e−R t0 ρ(s)dsσ(t)η(t)S(t)dW (t) .

Rewriting this into an integral equation on V (t) we get

e−R T0 ρ(s)dsV (T ) = V (0) +

∫ T

0e−

R t0 ρ(s)dsσ(t)η(t)S(t)dW (t) .

Applying the generalized Clark-Ocone theorem to G(ω) = e−R T0 ρ(s)dsF (ω) we have

G(ω) = EQ(G) +∫ T

0EQ

(DtG−G

∫ T

tDtθ(s)dW (s)|Ft

)dW (t) .


By uniqueness, from this it follows that

V (0) = EQ(G) (3.15)

and the required risky investment at time t is

η(t) = e−R t0 ρ(s)dsσ−1(t)S−1(t)EQ

(DtG−G

∫ T

tDtθ(s)W (t)|Ft

). (3.16)

3.3.2 Black-Scholes model

Let us assume now that ρ(t, ω) = ρ, µ(t, ω) = µ and σ(t, ω) = σ 6= 0 are constants. Then

θ(t, ω) = θ =µ− ρσ

is a constant and therefore Dtθ = 0. Hence

η(t) = eρ(t−T )σ−1S−1(t)EQ(DtF |Ft) . (3.17)

In the classical Black-Scholes model V (T, ω) = F (ω) represents the payoff of a (Euro-pean) call option and it is given as

F (ω) = (S(T, ω)−K)+ ,

where K is the exercise price and then V (0) is the price of the option. Clearly, this optiongives the owner the right to buy the stock with the value S(T ) at the fixed price K. Thusif S(T ) > K the owner of the option gets the profit S(T, ω) − K, otherwise the ownerdoes not exercise the option and the profit is 0. Since the function f(x) = (x−K)+ is notdifferentiable at x = K we cannot use the chain rule directly. Therefore, we approximatef with a series of functions fn ⊂ C1 with the property

fn(x) = f(x) for |x−K| ≥ 1n

and 0 ≤ f ′n(x) ≤ 1 for all x.Introducing Fn(ω) = fn(S(T, ω)) we get that

DtF (ω) = limn→∞

DtFn(ω) = 1[K,∞](S(T, ω))DtS(T, ω) = 1[K,∞](S(T, ω))S(T, ω)σ .

Hence, by (3.17)

η(t) = eρ(t−T )S−1(t)EQ(S(T )1[K,∞](S(T ))|Ft) .

By the Markov property of S(t) this is the same as

η(t) = eρ(t−T )S−1(t)EyQ(S(T − t)1[K,∞](S(T − t)))y=S(t) ,

where EyQ is the expectation when S(0) = y. The stochastic differential equation for S(t)can be rewritten as

dS(t) = µS(t)dt+ σS(t)dW (t)

= (µ− σθ)S(t)dt+ σS(t)dW (t)

= ρS(t)dt+ σS(t)dW (t) .


The solution of this linear stochastic differential equation is well known:

S(t) = S(0) exp((ρ− 12σ2)t+ σW (t)) .

Hence, we conclude that required risky investment is

η(t) = eρ(t−T )S−1(t)Ey(Y (T − t)1[K,∞](Y (T − t)))y=S(t) , (3.18)

whereY (t) = S(0) exp((ρ− 1

2σ2)t+ σW (t)) .

Since the distribution of W (t) is N(0, t), it is possible to express the solution (3.18) interms of quantities involving S(t) and the normal distribution.

As mentioned before, η(t) corresponds to the capital we must invest in the risky in-vestment at times t ≤ T in order to get the payoff F (ω) = (S(T, ω)−K)+ almost surely attime (maturity) T . The constant V (0) represents the corresponding initial fortune neededto achieve this. In other words, V (0) is the right price for such an option at time t = 0.Using (3.15) we get the famous Black-Scholes formula for this price:

V (0) = EQ(e−ρTF (ω)) = e−ρTEQ((S(T )−K)+) = e−ρTE((Y (T )−K)+) ,

which again can be expressed explicitly with the normal distribution function.

Remark. If ρ(t), µ(t) and σ(t) are Markov processes, i.e. when the price S(t) is given bya stochastic differential equation of the form

dS(t) = µ(S(t))S(t)dt+ σ(S(t))S(t)dW (t)

where µ : R→ R and σ : R→ R are given functions, then there is a well-known alternativemethod for finding the option price V (0) and the corresponding replicating portfolio η(t)using partial derivatives (greeks). However, this classical method does not work in thenon-Markovian case. The method based on the Clark-Ocone formula has the advantagethat it is valid in the more general, non-Markovian cases.

Chapter 4

Summary and conclusions

In this thesis we surveyed the most important concepts and results of Malliavin calculustogether with some interesting applications. In the introduction of the thesis the followingquestions has been asked as the main research question. We discuss them based on theresults of the thesis.

1. What are the key concepts and results in the recently developed mathematical fieldof Malliavin calculus?

The notions and theorems of the stochastic calculus of variations were summarizedin Chapter 1. The derivative operator D was introduced and extended to a widerange of random variables. It has been shown that it acts nicely on the Wienerchaoses. The special white noise setting has been investigated in more details. Inthis case, the adjoint operator of D, denoted by δ can be regarded as an extensionof the Ito integral. The Clark-Ocone formula specified the underlying process in theIto representation theorem for the case of nice random variables.

2. In which application areas has the theory been adapted for use?

From the many interesting applications developed in the recent years we collectedthree important results. Using on the Malliavin derivative matrix, a stochastic ver-sion of Hormander’s condition were given for the absolute continuity of the solutionsstochastic differential equations. With Ornstein-Uhlenbeck operators bounds on theKolmogorov distance of two random variables based on the Malliavin derivativeswere developed. Finally, the generalized Clarck-Ocone formula has been used todevelop a Black-Scholes formula for option pricing.

Nevertheless, the research is not yet finished. Several directions exists for future re-search. For example, there is a current trend in the discrete approximation of continuousstochastic concepts. The discrete approximation of the Malliavin derivate and the Wienerchaos can be useful in Monte Carlo algorithms or financial predictions. Another interestingdirection is the application of the theory in game theoretic market models.

To sum it up, the Malliavin calculus is an important and useful theory, which hasalready gained several applications and set up many open questions.

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Bibliography

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[3] P. Fitz. An introduction to Malliavin calculus. Lecture Notes, Courant Institute ofMathematical Sciences, New York University, 2005.

[4] L. Hormander. Hypoelliptic second order differential equations. Acta Mathematica,119(1):147–171, 1967.

[5] F.C. Klebaner. Introduction to stochastic calculus with applications. Imperial CollegePress, 2005.

[6] P. Malliavin. Stochastic calculus of variations and hypoelliptic operators. StochasticAnalysis, Editor K. Ito, Kinokuyina, Tokyo, pages 195–263, 1976.

[7] P. Malliavin. Ck-hypoellipticity with degeneracy. Stochastic analysis, pages 199–214,1978.

[8] I. Nourdin and G. Peccati. Stein’s method meets Malliavin calculus: a short surveywith new estimates. Arxiv preprint arXiv:0906.4419, 2009.

[9] D. Nualart. The Malliavin Calculus and Related Topics. Springer-VerlagBerlin/Heidelberg, 2006.

[10] B. Øksendal. An introduction to Malliavin calculus with applications to economics.Lecture Notes, Norwegian School of Economics and Business Administration, Norway,1997.

[11] D. Petz. Linear analysis (in Hungarian). Akademiai Kiado, 2002.

[12] V.S. Vladimirov. Methods of the theory of generalized functions. CRC, 2002.

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