A Perturbation Approach to Continuous-time Portfolio Selection∗

Dietmar P.J. LeisenUniversity of Mainz

Gutenberg School of Management and Economics55099 Mainz, Germany

Email: leisen@uni-mainz.de

This version: July 27, 2015

AbstractThis paper studies portfolio selection in continuous-time models with stochastic investment op-portunities. We consider asset allocation problems where preferences are specified as power util-ity derived from terminal wealth as well as consumption-savings problems with recursive utilityEpstein-Zin preferences. The paper introduces a new form of approximate dynamic program-ming (ADP) methods by perturbing the coefficients of the stochastic dynamics. We represent theHamilton-Jacobi-Bellman equation as a series of partial differential equations that can be solvediteratively in closed-form through computer algebra software, at any desired accuracy.

Keywordsperturbation, hedge demand, consumption, stochastic state variables

JEL ClassificationG11, G13

∗I thank Kenneth Judd and Kian Guan Lim, as well as participants of the Computation, Economics and FinanceWorkshop at the University of Zurich and of the Risk Management conference in Singapore for their comments andsuggestions.

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1 Introduction

Dynamic portfolio selection is a stochastic programming problem; closed-form solutions are known

only in special cases, and so many computational techniques have been developed. We focus on

so-called approximate dynamic programming (ADP) methods, i.e. techniques that approximate

the value function (a.k.a. cost-to-go function). While perturbation approaches have been ap-

plied successfully to many static stochastic optimization problems, a natural extension to dynamic

(continuous-time) problems is not available.

We develop a perturbation approach to study an important problem from finance, dynamic

portfolio selection under stochastic investment opportunities. To our knowledge, the literature has

studied exclusively problems where drift and volatility of the assets are time-independent; therefore,

we restrict ourselves to such specifications. Our approach focuses on a constant interest rate and a

bivariate diffusion process for the (joint) dynamics of the stock and the state-variable; it covers a

wide variety of financial asset dynamics in the literature, e.g. models with stochastic market prices

of risk and stochastic volatility models1. We consider asset allocation problems where the agent

optimizes utility derived from terminal wealth and consider consumption-savings problems where

the agent optimizes Epstein-Zin recursive utility preferences. (This includes time-separable CRRA

preferences as a special case.)

Our approach perturbs the underlying diffusion coefficients in such a way that the Hamilton-

Jacobi-Bellman (HJB) equation of the value function can be separated into an infinite sequence of

partial differential equations (PDEs) and that these PDEs have an iterative structure. We solve

the PDEs iteratively; this amounts to finding the terms in a functional series approximations of

the value function; calculating the approximation of any order, say k, involves a time-integration

of sums of functional terms that are (non-negative) integer powers in time; the summation terms

are derivatives in the state variable of lower order functional series terms (1, . . . , k − 1). Due to

this particular structure, time-integration is straightforward and all iterative steps can be solved in

closed form through computer algebra software. We prove that the (functional) series of solutions

converges to the value function of the underlying dynamic optimization problem. In addition, we

implement our perturbation approach to study portfolio selection in the main model for stochastic

market prices of risk (Kim and Omberg (1996)) and discuss its numerical performance in terms of

accuracy.

ADP applications in the OR/MS field focus on so-called approximate linear programming

(ALP), (Schweitzer and Seidmann (1985), De Farias and Van Roy (2003)); in the finance liter-

ature, regression methods have been studied to price American options (Longstaff and Schwartz

(2001), Tsitsiklis and Van Roy (2001)) and portfolio selection (Brandt et al. (2005)). In addition

1Our process dynamics covers a wide range of dynamics of the state variable and some of these lead to unboundedprocesses. However, unbounded volatility and market price of risk are unrealistic from an empirical perspective.As the unbounded process creates technical problems, we replace the process at times by a bounded one for whichdrift/volatility functions are identical to the original functions on a subset. For consistency, we only discuss theportfolio selection problem of state variable realizations where drift/volatility of the original process coincides withthat of the (bounded) replacement process.

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to ADP methods, the finance and economics literature studied a variety of methods: Complete

markets permit the use of martingale methods to restate portfolio selection problems as a static

problem (Cox and Huang (1989)) or to use simulation methods (Detemple et al. (2003), Detemple

and Rindisbacher (2010)). Duality theory has been applied to American option pricing (Haugh

and Kogan (2004)), investing under portfolio constraints (He and Pearson (1991), Cuoco (1997)),

investing under illiquidity (Schwartz and Tebaldi (2006)) and robust decision making (Lim et al.

(2011), Lim and Shanthikumar (2014)). Computational methods include the log-linearization of

the consumption-wealth ratio (Campbell and Viceira (1999), Chacko and Viceira (2005)), Markov-

chain approximations (Kushner and Dupuis (1992), Munk (2000)), Feynman-Kac fixed-point meth-

ods (Kraft et al. (2014)), finite difference methods (Brennan et al. (1997)), perturbations of the

utility function (Kogan and Uppal (2001)) and perturbations of option pricing formulas (Fouque

et al. (2000)).

Our method provides closed-form approximations at any desired accuracy. This is a major

advantage over current computational methods to dynamic portfolio selection problems, similar to

the (single-period) small-noise expansion of Samuelson (1970), see also Judd (1996). Schwartz and

Tebaldi (2006) study a related problem, investing with an illiquid asset in a classical Merton setup;

they provide an ADP based on an ad-hoc power series expansion of the dual problem and solve it

iteratively. Differently to them, however, we start from a perturbation of the diffusion coefficients;

this leads us to expansion terms that may be different from power terms. Our closed-form expres-

sions can be used for a variety of economic analysis: for example, they serve for comparative statics

in a straightforward way; they may also be used easily for cross-model comparisons of the resulting

portfolio selection implications. Such analysis may provide a better understanding of the models

underlying portfolio selection as well as of the trade-offs in modeling choices.

The most important contribution lies in the computational efficiency of our method compared

others. It provides a series of (closed-form) functional approximations of the parameters of interest

and at any desired accuracy, that can be set up efficiently in computer algebra software: each

iterative step involves the time-integral of summations over derivatives of lower order approxima-

tions. (Computer algebra software determines easily the sums of derivatives but may potentially

encounter difficulties with (time-)integration; yet, we show that the time-dependency of all terms

shows up as powers of time and can be carried out easily.) It is also important to note that over

the lifespan of an optimization problem, we need to determine the functional approximation only

once (at the beginning): at any point in time, the parameters of interest (value function, portfolio

selection) can then be evaluated in a computationally efficient way by plugging current time and

state-variable into our functional approximation.

The remainder of the paper is organized as follows. The next section presents the setup of the

portfolio selection problem: the continuous-time process dynamics of the stock and state variable as

well as the preference structure. Section 3 introduces our general perturbation approach based on

perturbations of of coefficients in the dynamics of economic variables; the following section links this

back to the original problem of interest and characterizes the resulting (iterative) approximations.

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Section 5 focuses on a stochastic market price of risk to evaluate the numerical performance of our

perturbation approach. Section 6 concludes. All proofs are postponed to the appendix.

2 The Portfolio Selection Problem

2.1 Stochastic Investment Opportunities

We fix a time-horizon T , as well as a continuous-time interval [0, T ], with the understanding that

time 0 is today and assume a probability space (Ω,F , P ), where P denotes the probability measure.

Riskfree investing and borrowing is possible at the instantaneous rate r. A risky stock is traded

continuously in the market; the dynamics of the stock price S and the state variable X takes values

in R+ × Ξ ⊂ R2 and is given on the time interval [0, T ] through the joint stochastic differential

equation2

dSt = (r + λS(Xt))Stdt+ σS(Xt)StdB1t, (1)

dXt = µX(Xt)dt+ σX(Xt)dB2t. (2)

Here, λS , σS , µX , σX are real-valued functions of the state-process X, and (B1, B2) is a bivariate

Brownian motion with constant correlation ρ. In applications, we are particularly interested in the

standard parametrization, where the drift and volatility functions fulfill for x ∈ Ξ:

λS(x) = λxν , σS(x) = σSxβ1 , µX(x) = κxβ2(x− x), σX(x) = σXx

β3 . (3)

It is a common assumption that the state variable process is mean-reverting; the drift function

µX in equation (3) allows for this by setting x = 0; then x is called the long-run mean and κ the

mean reversion rate. For technical reasons we impose the restriction ν ≥ β1 throughout this paper.

Equation (3) covers a wide variety of common models with stochastic investment opportunities.

From a technical viewpoint, it covers the (special) functional forms of drift and volatility, for

which affine and quadratic affine solutions to the portfolio selection problem exist (Liu (2007)).

Moreover, it covers major models that have been studied in the literature. For example, we can

study stochastic market prices of risk, setting ν = 1, σS = λ and β1 = β2 = β3 = 0; this setup has

been suggested by Kim and Omberg (1996); later, Wachter (2002) studied it with the restriction

ρ = −1. In addition, we can study a variety of stochastic volatility models. Equation (3) covers

the stochastic volatility models of Christoffersen et al. (2010) by setting ν = β2 + 1, σS = 1, and

2We introduce our method for a single risky asset, a single state variable and a constant interest rate; this leadsto an HJB equation that splits into an operator equation in time and the state-variable and is particularly easy todiscuss. Applying it to problems with multivariate processes of state-variables and/or multiple risky assets and/ornon-homothetic preferences, the HJB equation would lead to an operator equation in time, state-variables (includinginterest rate) and potentially wealth. Applying it to infinite horizon problems would lead to an operator equationin the state-variable, only. While our approach would easily provide a sequence of functional series approximations,the main challenge in such applications would be to provide a verification theorem as well as a theorem that ensuresconvergence. A generalization to time-dependent drift and volatility functions would also be feasible as long as theiterative time-integration procedure can be carried out at all steps.

4

β1 = 12 , and β2 = 0, 1 and β3 = 1

2 , 1,32 . Equation (3) also includes the common Heston (1993)

model by setting ν = 12 or ν = 1 and σS = 1, β1 =

12 , β2 = 0, β3 =

12 , as well as the Hull and White

(1987) model by setting ν = 12 , σS = 1, β1 = 1

2 , β2 = 0, β3 = 1. In addition, the Stein and Stein

(1991) and Scott (1987) model are covered by setting ν = 1, σS = 1, β1 = 1, β2 = 0, β3 = 0.

Some of our state process descriptions exceed any lower and any upper bound with non-zero

probability: For example, the state process in Kim and Omberg (1996), Stein and Stein (1991),

and Scott (1987) represents a so-called Ornstein-Uhlenbeck process; it is well known that paths of

such processes are unbounded. These modeling issues have been largely neglected in the literature3.

Empirically, very large levels for volatility or market prices of risk are unknown; similarly, negative

levels are questionable from a theoretical perspective.

In addition to the empirical observation that state variables are bounded, it is also convenient

to ensure this from a technical perspective4 and therefore we assume this right from the start.

To circumvent the issue that some process descriptions are not bounded, we intend to adopt the

functional forms of equation (3) on a subset of Ξ, only. To define this, we adopt numbers 0 ≤ xL ≤xl < xu ≤ xU <∞ and set Ξ = [xL, xU ]:

Assumption 1 Given are (four) constants ν ≥ β1, β2, β3 that are non-negative integer multiples

of 1/2, and λ, σS , κ, x, σX that are non-negative constants. The drift and volatility functions fulfill

equation (3) for x ∈ [xl, xu].

To ensure that the state variable only adopts values in the bounded interval Ξ = [xL, xU ], we

adopt:

Assumption 2 Fix xL < y0 < xU . For xL < y < xU we define the lower and upper scale densities

sL, sU by setting:

sL(y) = exp

(−∫ y

y0

µX(y)

σ2X(y)dy

), and sU (y) = exp

(−∫ y0

y

µX(y)

σ2X(y)dy

).

We assume that the drift and volatility functions λS , σS , µX , σX as well as the ratios λSσS, σXσS are

all (well-defined real-valued and) infinitely differentiable functions on Ξ. Finally, we assume for

xL < x0 < xU that

limx↓xL

∫ x0

xsL(y)dy = ∞, and lim

x↑xU

∫ x

x0

sU (y)dy = ∞. (4)

The scale conditions (4) are similar to an asymptotic at the left/right boundary: essentially,

they require that getting to the boundary, the volatility of the state variable tends to 0 sufficiently

3A notable exception is Detemple et al. (2003) for the interest rate (short rate) process: they provide an explicitalternative process that depends functionally on the entire range of validity on the exogenously specified (lower andupper) bounds.

4At first, such a restriction may appear odd. However, note that this is a typical procedure in methods that solvethe resulting Hamilton-Jacobi-Bellman equation (or its associated PDE that characterizes the functional impact ofstate variables) through finite difference methods, see, e.g. Kraft et al. (2014). Any such grid is bounded, but theassociated restrictions are imposed when it comes to solving the PDE, only. We impose this right from the start,here.

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fast, in order that the drift pulls the process away from the boundary. It is well known that a

solution to the SDE (1, 2) exists, if we impose growth and Lipschitz conditions on the drift and

volatility functions, see, e.g. Revuz and Yor (1999); infinitely differentiable drift and volatility

functions fulfill growth and Lipschitz conditions. Using the scale condition together with results in

Karlin and Taylor (1981), we find:

Proposition 3 Under assumption 2, a unique strong solution (S,X) exists to the joint dynamics

in equations (1, 2) that has continuous paths with probability 1 and stays within R+ × Ξ, i.e.

P [St ≥ 0 and Xt ∈ Ξ for all 0 ≤ t ≤ T ] = 1.

In applications we first adopt assumption 1 on the strict subset [xl, xu] ⊂ Ξ; the ratios λSσS, σXσS

are then all (well-defined and) infinitely differentiable on the subset [xl, xu] ⊂ Ξ. In a second step

we then extend the functions to the entire interval Ξ using a suitable smooth pasting procedure in

such a way that assumption 2 holds. Extending the drift and volatility functions on Ξ that fulfill

the scale condition (4) is necessary to study a well-defined stochastic process for the state variable;

this extension does not affect our perturbation approach: starting with subsection 3.4 below we

will only study the subinterval [xl, xu].

2.2 Trading, Preferences and the Wealth Process

Throughout this paper, we study two related portfolio selection problems of an investor: (1)

consumption-savings portfolio choice (or intertemporal consumption and portfolio choice); (2) asset

allocation (or dynamic portfolio choice). To describe these two portfolio selection problems in a

unified way, we introduce parameters 0 < γ = 1, ε > 0 and the indicator variable χ; we set χ equal

to 0 in the asset allocation problem and equal to 1 in the consumption-savings portfolio choice

problem. For later reference we note that

exp(χ ln ε) =

1 asset allocation problem

ε consumption-savings portfolio choice problem.

In line with Merton (1971) we assume perfect capital markets: there are no transactions costs;

the stock is infinitely divisible and short sale with full use of proceeds is allowed. Trading in the

stock takes place continuously in time, and we denote by ω = (ωt)0≤t≤T the so-called portfolio

weight process. Here, ωt denotes the relative weight of the time 0 ≤ t ≤ T investment in the stock;

the remainder 1−ωt is invested at the riskfree rate. (As usual, negative values for 1−ωt should be

interpreted as borrowing at the rate r.) We denote by C = (Ct)t the agent’s consumption process

financed by the investment strategy ω. Throughout, we view portfolio weight ω and consumption

C as functions of current time t, state-variable x and potentially current wealth w; we do not write

out these dependences explicitly to simplify notation. Then, the agent’s budget constraint is given

through the dynamics of the wealth process

dWt = (rWt + ωtλS(t,Xt)Wt − χCt) dt+ ωtσS(t,Xt)WtdB1t, (5)

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together with the dynamics of the state-variable X, equation (2). We require that the joint process

(ω,C) is chosen from the set of admissible controls

A = (ω,C)| wealth dynamics(5) has a unique solution;Ct and Wt ≥ 0 for 0 ≤ t ≤ T ;CT =WT .

Our goal is to determine at all times 0 ≤ t ≤ T :

J(t, w, x) = maxω,C

E

[∫ T

tχf(Cs, J(s,Ws, Xs))ds+ exp(χ ln ε)

W 1−γT

1− γ

∣∣∣∣∣Wt = w,Xt = x

], (6)

where the wealth process is given by (5) and maximization is taken over the remaining time of the

weight ω and consumption C. The function J is often called the continuation utility or the value

function (a.k.a. cost-to-go function); throughout, we refer to it as the indirect utility function, for

reasons that will be detailed below.

This maximization problem (6) allows us to study both portfolio selection problems in a unified

framework. The case χ = 0 corresponds to the problem of a price taking agent who maximizes utility

derived from terminal wealth WT , as in Merton (1971). In line with the literature we consider a so-

called utility function with constant relative risk aversion (CRRA, a.k.a. power utility); 0 < γ = 1

denotes the (relative) risk aversion coefficient. (The case γ = 1 would correspond to an agent

with logarithmic utility; while this case could be studied, we refrain from doing so to simplify

our exposition.) It is common in the asset allocation literature to refer to J(t, w, x) as the indirect

utility function and we follow that convention. Overall, the case χ = 0 allows us to study a classical

asset allocation problem, i.e. a portfolio choice problem without consumption.

The other case is χ = 1. We study the problem setup in Kraft et al. (2013) and adopt their

notation; in particular, we denote by δ > 0 the rate of time preference, by ψ > 0 the elasticity of

intertemporal substitution, and by f the so-called normalized aggregator of current consumption

and so-called continuation utility. The case ψ = 1 has been studied in the literature, see, e.g. Kraft

et al. (2013) for a discussion; to simplify our exposition we assume ψ = 1 throughout this paper.

For a deeper introduction into Epstein-Zin preferences, and a discussion thereof, we refer the reader

to Chacko and Viceira (2005) and Kraft et al. (2013), among others.

At this stage it is only important for us to recall that this allows us to set the elasticity of

intertemporal substitution (ψ > 0) separately from the (relative) risk aversion coefficient (0 < γ =1). Moreover we recall that ψ = 1/γ reduces the recursive utility consumption-savings problem to

the common consumption savings problem with time-separable CRRA preferences and (relative)

risk aversion coefficient γ. This restriction will be imposed at times to compare results with those

of the so-called Merton consumption-savings problem. Leaving aside the multiplicative term in ε,

our optimization problem matches exactly that in Duffie and Epstein (1992) of a price taking agent

with stochastic differential utility derived from lifetime consumption, a continuous-time version of

the so-called Epstein and Zin (1989) preferences.

Throughout this paper, we do allow explicitly for a bequest function εW1−γ

1−γ ; the parameter ε > 0

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allows us to adjust the relative importance of bequest and lifetime consumption; this is similar to

Liu (2007) in his analysis of portfolio choice with CRRA preferences. Throughout, the presentation

of our perturbation approach is based on a logarithmic transformation that we introduce in the

next subsection; this transformation simplifies our presentation but requires ε > 0. (Our approach

could be implemented without a logarithmic transformation to cover the case ε = 0, but this

complicates the implementation and presentation.) Thus, the case χ = 1 describes the portfolio

selection problem of a price taking agent who derives utility from consumption, i.e. a general setup

to study consumption-savings portfolio choice problems.

We assumed previously ψ = 1 and define

θ =1− γ

1− 1ψ

and η = −ψθ=

1− ψ

1− γ. (7)

Assumption 4 Assume one of the following (partially exclusive) assumptions on preference pa-

rameters hold: (a) γ > 1, ψ > 1, (b) γ > 1, ψ < 1 with γψ ≤ 1, (c) γ < 1, ψ < 1, (d) γ < 1, ψ > 1

with γψ ≥ 1.

This assumption is identical to the condition in Theorem 3.1 of Kraft et al. (2013). It will be

used later in the verification theorem part of our Theorem 7.

3 The Portfolio Selection Problem in Perturbed Economies

Perturbation approaches introduce an additional parameter, here denoted p; this paper perturbs

the underlying drift (and volatility) functions and perturbs the preference structure. Throughout,

we study only economies with p > 0.

3.1 Differences to Common Perturbation Approaches

In economics and finance, decision variables are often the solution to an appropriately defined

functional equation. Perturbation approaches typically invoke a form of the implicit function the-

orem (IFT) that serves two goals, see Judd (1998): first, successive implicit differentiation permits

iterative calculation of the series expansion coefficients for decision variables; second, the IFT is

used to prove convergence of the series in some neighborhood of p = 0. Financial applications of

this procedure are given by Judd and Guu (2001), and Judd and Leisen (2010), among others; for

economic applications, see, e.g. Jin and Judd (2002).

Although our approach shares the spirit of perturbation approaches it is important to stress

differences to the standard method: first of all, previous studies focused on single-period or discrete-

time multi-period problems and, typically, solve a functional equation for a single real number or

a multidimensional real vector as a decision variable. However, we consider a continuous-time

problem that we re-express using a functional operator on a space of (sufficiently differentiable)

functions; throughout we solve for a sequence of functions.

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Second, in previous perturbation approaches with risky economic/financial variables, the noise

term is often scaled proportionally by the perturbation parameter. It is typically assumed that

p is “small,” i.e. “close” to zero; this relates to small-noise expansions, introduced by Samuelson

(1970). However, serious technical issues need to be resolved, when the p = 0 economy corresponds

to an economy where all risk vanishes and the stock becomes indistinguishable from the riskfree

investment (so-called trivial economy). In our approach, however, the perturbation parameter does

not have to scale noise (proportionally) as our ultimate goal is solving a (portfolio selection) problem

for p = 1. Our goal also means that we are not interested in the neighborhood of p = 0, we are

only interested in a neighborhood of p = 1. It is crucial to ensure a strictly positive convergence

radius for our functional series expansion and we do so in theorem 10 at the end of this section.

Third, we do not use implicit differentiation to determine the expansion terms. Instead we follow

Chabi-Yo et al. (2014) in their analysis of decision variables: they write the underlying problem as

a series of problems to solve, one for each expansion order and match coefficients. Our perturbation

approach here is insofar different from theirs as they look for a solution to a real valued function,

while we looked for a solution to an operator on a function space.

3.2 Setting up the Perturbation

We consider a sequence of economies indexed by a parameter p > 0. In each p-economy, we assume

that the stock price Sp with state variable(s) Xp follows the joint dynamics

dSp,t = (rp + λS,p(Xp,t))Sp,tdt+ σS,p(Xp,t)Sp,tdB1t, (8)

dXp,t = µX,p(Xp,t)dt+ σX,p(Xp,t)dB2t. (9)

This joint stochastic differential equation is analogous to (1, 2); the difference is that here the

interest rate r and the functions λS , σS , µX , σX depend on the parameter p. (We use the “original”

bivariate Brownian motion; in particular we leave the correlation ρ independent of p.) For simplicity,

and unless necessary to prevent confusion, we usually do not write out explicitly the dependence

of λS,p, σS,p, µX,p, σX,p on the current state-variable.

In each p-economy we assume perfect capital markets with continuous trading (analogous sub-

section 2.1) and denote by ωp, Cp the process of wealth weights and of consumption. This leads in

each p-economy to the wealth process Wp that generalizes equation (5):

dWp,t = (rWp,t + ωtλS(Xp,t)Wp,t − χpCp,t) dt+ ωtσS(Xp,t)Wp,tdB1t, (10)

We require that the joint process (ω,C) is chosen from the set of admissible controls Ap =

(ω,C)| wealth dynamics (10) has a unique solution;Cp,t and Wp,t ≥ 0 for 0 ≤ t ≤ T ;Cp,T =Wp,T .We replace in equation (6) the indirect utility functions J and the processes X,W,ω,C with their

p-economy counterparts Jp, Xp,Wp, ωp, Cp; in a generalization of equation (6), our goal is to deter-

9

mine at all times 0 ≤ t ≤ T :

Jp(t, w, x) (11)

= maxωp,Cp

E

[∫ T

tχpf(Cp,s, Jp(s,Wp,s, Xp,s)ds+ exp(χ ln ε)

W 1−γp,T

1− γ

∣∣∣∣∣Wp,t = w,Xp,t = x

].

Note that we replace χf in the original problem by χpf = pχf but leave exp(χ ln ε) unchanged.

The dependence on χp in the first term means that we perturb the preferences, in addition to the

process structure, see above. Leaving exp(χ ln ε) unchanged is convenient in later analysis where we

study PDEs parametrized by the perturbation parameter p. (Essentially, we set up our perturbation

in such a way that the boundary conditions are independent of p.) Since we are only interested in

the portfolio selection problem for the p = 1 economy, nothing prevents us from varying χp in a

p-economy, as long as we ensure χ1 = 1.

There is great freedom in choosing the functional form of the perturbation; ultimately, the

functional dependence of rp, λS,p, σS,p, µX,p, σX,p on p characterizes the perturbation approach. The

choice impacts the implementation and our presentation of it; this paper focuses on the perturba-

tion:

Assumption 5 In the p-economy, the economic variables are given by

χp = pχ; rp = pr;λS,p = pλS ;σS,p =√pσS ;µX,p = pµX ;σX,p =

√pσX . (12)

We expand the drift functions (and analogously the interest rate) by p and volatility functions by√p; this is the continuous-time analogue to the (static, classical) small-noise expansion approach of

Samuelson (1970) and Judd (1996), where volatility is scaled by the square root of time (increments)

and expectation scales linearly in time (increments). Since χ1 = χ, r1 = r, λS,1 = λS , σS,1 =

σS , µX,1 = µX , σX,1 = σX , the stochastic processes and optimization problem (11) for p = 1 are

identical to our earlier stochastic processes (1, 2) and optimization problem (6); in particular, we

have the preference structure in our original economy of interest for p = 1, such that the p = 1

economy will give us the optimal allocation of interest.

Proposition 6 Assume p > 0 and assumptions 2, 5 hold. Then a unique strong solution (Sp, Xp)

exists to the joint dynamics in equations (8, 9) that has continuous paths with probability 1 and

stays within R+ × Ξ, i.e. P [Sp,t ≥ 0 and Xp,t ∈ Ξ for all 0 ≤ t ≤ T ] = 1.

3.3 Expressing the Optimization Problem through a Sequence of PDEs

The principle of optimality, see, e.g. Pennacchi (2008), leads us to the Hamilton-Jacobi-Bellman

equation (henceforth HJB equation) in the asset allocation problem; similarly, Duffie and Epstein

(1992), Chacko and Viceira (2005) and Kraft et al. (2013) exhibit the corresponding HJB equation

for the consumption savings problem.

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For later reference we define C1,2([0, T ] × Ξ), the space of functions [0, T ] × Ξ → R that are

continuously differentiable in time, and twice differentiable in space. We view portfolio weight ωp

and consumption Cp as functions of current time 0 ≤ t ≤ T , state-variable x and potentially current

wealth w. It is common to conjecture that the indirect utility function is separable into the product

of a time/state function and a wealth function. We follow that procedure and conjecture that

Jp(t, w, x) =w1−γ

1− γexp (gp(t, x)) . (13)

Theorem 7 Assume p > 0 and assumptions 2, 4 & 5. Denote gp ∈ C1,2([0, T ]×Ξ) the solution of

∂gp∂t

= χp

(−δ

ψ

ηexp (ηgp) + δθ

)+ rp(γ − 1) +

γ − 1

2γ

(λS,pσS,p

)2

(14)

+

(γ − 1

γρλS,pσS,p

σX,p − µX,p

)∂gp∂x

+γ(ρ2 − 1)− ρ2

2γσ2X,p

(∂gp∂x

)2

−σ2X,p2

∂2gp∂x2

,

subject to the boundary condition

gp(T, ·) = χ ln ε. (15)

Define functions of current time 0 ≤ t ≤ T and current state-variable x ∈ Ξ:

ωp =λS,pγσ2S,p

+ ϕp, where ϕp =ρσX,pγσS,p

∂gp∂x

; cp = δψ exp (ηgp) . (16)

Assume that cp = δψ exp(ηgp) < 1 on [0, T ]×Ξ. Then (ωp, cp) are admissible strategies that describe

the optimal wealth weights ωp and the optimal consumption wealth ratio cp (in consumption-savings

problems).

Our proof is provided in the appendix and uses the verification Theorems 3.1 and 3.2 in Kraft

et al. (2013); in particular, the appendix proves their local martingale condition; essentially this

holds in our setup since paths of the state process are bounded, see assumption 2. The condition

cp < 1 is a sufficient one to ensure a strictly positive wealth process; it is analogous Kraft et al.

(2013), see, e.g., their proposition C.6, and can easily be checked in applications.

Typically, it would be conjectured that Jp(t, w, x) =w1−γ

1−γ gζp for a suitable constant ζ and func-

tion gp of time and the state variable, see, e.g. Kraft et al. (2013); then, gp would be characterized

by a PDE that is structurally different from (14), and demand would be driven by∂ ln gp∂x = 1

gp

∂gp∂x

instead of∂gp∂x (with our conjecture). The perturbation approach provides approximations of the

parameter of interest through suitable base functions and this feeds well into equation (16); as such

it is a matter of convenience.

Proposition 7 decomposes the optimal investment strategy ω into two terms: λS,p/(γσ2S,p) is

the so-called myopic investment demand; it reflects the risk-return trade-off for current investment

opportunities. Our focus is on ϕp, the so-called (intertemporal) hedge demand, that displays how

the agent reacts to possible changes in the investment opportunity set.

11

3.4 The Operator Problem

Theorem 7 characterizes the portfolio selection problem through solutions to the PDE (14, 15)

that are defined on the entire set [0, T ] × Ξ. We recall that our focus is on the portfolio selection

problem for x ∈ [xl, xu] ⊂ Ξ. While it was inevitable to study state variables that take value

in the entire set Ξ so far, throughout the remainder of this paper, we study the PDE (14) with

boundary condition (15) on the set [0, T ]× [xl, xu], only: in particular, we intend to characterize the

logarithmic transformation function gp, hedge demand ϕp and (if applicable of) consumption-wealth

ratio cp on this subset [xl, xu], only.

For later reference we define we the set I = [0, T ] × [xl, xu], the space C1,2(I) of functions

I → R that are continuously differentiable in time t, and twice differentiable in x, as well as the

space G = g ∈ C1,2(I)|g(T, ·) = 0. Also, we define by || · ||[xl,xu] the maximum of continuous

functions on the (compact) interval [xl, xu] and by || · ||∞ the maximum on the (compact) set I,i.e. for f1 ∈ C([xl, xu]) and for f2 ∈ C(I) we set ||f1||[xl,xu] = maxx∈[xl,xu] |f1(x)|, and ||f2||∞ =

max(t,x)∈I |f2(t, x)|; we define the norm || · ||G on G by setting ||g||G = ||g||∞ + ||gt||∞ + ||gx||∞ +

||gxx||∞ for g ∈ G. Note that both G with norm || · ||G and C(I) with norm || · ||∞ are Banach

spaces. Finally, we define functions ξi : [xl, xu] → R for i = 1, 2, 3, 4, by setting for xl ≤ x ≤ xu:

ξ1(x) = r(γ − 1) +γ − 1

2γ

(λS(x)

σS(x)

)2

, ξ2(x) =γ − 1

γρλS(x)

σX(x)

σS(x)− µX(x), (17)

ξ3(x) =γ(ρ2 − 1)− ρ2

2γσ2X(x), and ξ4(x) = −

σ2X2

(x). (18)

Note that the functions ξi (i = 1, 2, 3, 4) are all well-defined under assumption 2. We stress that they

depend only on the state variable x; to simplify exposition here and throughout the remainder of

this paper we do not write out this dependence explicitly. We then define two functional operators

L,G : C1,2(I) → C(I) by setting for f ∈ C1,2(I):

Lf =∂f

∂t, and Gf = ξ1 + ξ2

∂f

∂x+ ξ3

(∂f

∂x

)2

+ ξ4∂2f

∂x2+ χ

(−δ

ψ

ηεη exp(ηf) + δθ

). (19)

We calculate χεχη = χεη and use this to re-express the PDE in Proposition 7 in terms of an operator

equation that we will study in the remainder of this paper:

Theorem 8 5 Adopt assumption 5, fix a p > 0 and study gp = gp + χ ln ε. Solving the PDE (14),

subject to the boundary condition (15) is equivalent to solving

Lgp = p · G(gp), subject to the boundary condition gp(T, ·) = 0. (20)

Alternatively, we may think of this as solving for gp ∈ G with Lgp = p · Ggp.5The transformation from gp to gp (and vice versa) substitutes the original boundary condition (15) on gp by the

boundary condition on gp in equation (20). If this would be studied for all p ≥ 0 (including p = 0), the operatorequation could be interpreted as a homotopy deformation and studied through the so-called homotopy perturbationmethod, see He (1999).

12

It is critical to note that our perturbation is set up in such a way that the operator equation

in Theorem 8 can be written using two operators (L, G) that are independent of the perturbation

parameter p. Throughout we use the equivalence gp = gp + χ ln ε.

3.5 Iterative Procedure

This subsection describes our constructive approach of solving iteratively for the functional expan-

sion terms of gp. Throughout this subsection we look for analytic solutions to the operator equation

(20) for p in a suitable neighborhood of p = 1, i.e. we look for a 0 < p < 1 and a suitable series of

functions (gk)k such that for all |p− 1| < p

gp =∞∑k=1

pkgk, or equivalently gp = χ ln ε+∞∑k=1

pkgk, (21)

For future reference we define a function c0 and for k = 1, 2, . . . functions ck, ϕk by setting

c0 = δψεη, ck = δψεηk∑

n=1

ηn

n!

∑j1,...,jn≥1j1+...+jn=k

n∏m=1

gjm , and ϕk =ρσXγσS

∂gk∂x

. (22)

(All these functions gk, ϕk, ck depend on current time t and state-variable x; throughout this paper

we usually do not write out these dependences to keep notation simple.) Also, for any integer

k > 1, we define the operator Gk by setting, for f = (f1, . . . , fk) with fi ∈ C1,2(I), i = 1, . . . , k:

Gkf = ξ2∂fk∂x

+ ξ3

k−1∑j=1

∂fj∂x

∂fk−j∂x

+ ξ4∂2fk∂x2

− χδψεηk∑

n=1

ηn−1

n!

∑j1,...,jn≥1j1+...+jn=k

n∏m=1

fjm . (23)

Proposition 9 Assume a suitable 0 < p < 1 such that gp is analytic for all |p − 1| < p. Then,

the optimal hedge demand ϕp in the p-economy is analytic for all |p− 1| < p and, in consumption-

savings problems (χ = 1), the consumption-wealth ratio cp in the p-economy is analytic for all

|p− 1| < p. We have

ϕp =∞∑k=1

pkϕk, and cp = c0 +∞∑k=1

pk ck. (24)

In the remainder of this subsection we derive first the expansion terms iteratively using the

principle of matching “coefficients” (here: functions) in p; at the end of the subsection we provide

in Theorem 10 sufficient conditions that ensure that the (constructed) sequence converges in a

neighborhood of p = 1.

Using the principle of matching terms, we now derive a sequence of operator equations. For

this we note first that χ δψ

η εη exp(ηgp) = χ δ

ψ

η εχη exp(ηgp) = χ δ

ψ

η exp(ηgp) = χcp/η. Then, we plug

13

our functional series expansion of gp (see equation (21)) into the operator equation (20):

∞∑k=1

pkLgk = L(gp) = pG(gp)

= p

ξ1 + ξ2

∞∑k=1

pk∂gk∂x

+ ξ3

( ∞∑k=1

pk∂gk∂x

)2

+ ξ4

∞∑k=1

pk∂2gk∂x2

+ pχ

(−cpη

+ δθ

).

Next, we plug the resulting expansion based on (22, 24) into this equation; also, we note that(∂gp∂x

)2=(∑∞

k=1 pk ∂gk∂x

)2=∑∞

k=1 pk∑k−1

j=1∂gj∂x

∂gk−j∂x , with the understanding that summation over

an empty set is equal to zero. Collecting terms of same order in p, we get

∞∑k=1

pkLgk = p

(ξ1 + χ

(δθ +

δψ

ηεη))

+∞∑k=2

pkGk−1(g1, . . . , gk−1)

using the sequence of operators (Gk)k=1,2,.... Matching the terms of all orders k = 1, . . . in p, we

find the following sequence of operator equations to be solved for g1, gk+1 ∈ C1,2(I), one for each

expansion term k = 1, 2, . . .:

Lg1 = ξ1 + χ

(δθ − δψ

ηεη), and Lgk+1 = Gk(g1, . . . , gk), (25)

with boundary condition

gk(T, x) = 0, for all k = 1, 2, . . . and xl ≤ x ≤ xu. (26)

(The boundary condition (20) for any p becomes a sequence of boundary conditions, one for every

order k = 1, 2, . . . of pk.) This defines our iterative approach: note that any iteration step only

requires the information about lower order functional expansion terms.

Theorem 10 Assume 0 < ||ξi||[xl,xu] <∞ for i = 1, 2, 3, 4. In consumption-savings problems with

η > 0 also assume that − δψ

η εη exp(η) − δθ > 2||ξ1||[xl,xu]. Then, a number 0 < p < 1 exists such

that for every |p − 1| < p, equation (20) has exactly one analytic solution gp ∈ G. The series of

functions (gk)k ⊂ C1,2(I) in equation (21) can be constructed iteratively solving equations (25, 26).

This Theorem uses a suitable majorant to show convergence of the sequence we constructed

above. It ensures for our functional series expansion a positive convergence radius around p = 1;

the principle of matching coefficients only requires convergence in such a neighborhood and thus

allows us to determine the expansion terms iteratively (uniqueness). Together with the sequence

of boundary conditions (26) this means that we look in the remainder of this paper for expansion

terms in gk ∈ G for all k = 1, 2, . . .. Two comments are in order: First, it is important to note

that our theorem only provides a sufficient for convergence. Second, this theorem does not rely

on assumption 5 directly, but it will be applied together with theorem 8 which relies on that

14

assumption.

4 Approximation for the Intended Portfolio Selection Problem

The previous section introduced our perturbation procedure and characterized the functional series

expansion in p. This section explains how this allows us to study the intended portfolio problem

of section 2 and discusses properties of our approximation. Throughout, we look exclusively at the

interval [xl, xu] for the state variable. Also, we adopt the standard parametrization in assumption

1. We then calculate under assumption 1 that the functions ξi (i = 1, 2, 3, 4), defined in equations

(17, 18) are for xl ≤ x ≤ xu:

ξ1(x) = r(γ − 1) +γ − 1

2γ

λ2

σ2Sx2ν−2β1 , ξ2(x) =

γ − 1

γ

ρλσXσS

xν+β3−β1 − κxβ2(x− x), (27)

ξ3(x) =γ(ρ2 − 1)− ρ2

2γσ2Xx

2β3 , and ξ4(x) = − σX2x2β3 . (28)

We recall that our standard parametrization (see assumption 1) assumed ν ≥ β1 and note that

all these functions ξi (i = 1, 2, 3, 4) are continuous and bounded on [xl, xu]. For consumption-

savings problems it is cumbersome to characterize the sufficient condition in Theorem 10 and not

illustrative at this stage; section 5 will address the validity of this sufficient condition in concrete

applications.

4.1 Properties

The perturbation method gives solutions gp for the perturbation parameter within a suitable con-

vergence radius. This is necessary to write an iterative procedure but we will only interpret the

p = 1 economy: the stochastic differential equation (8, 9) and the associated optimization problem

(11) for the p = 1 economy match exactly our original problem of interest (1, 2, 6); this guarantees

that ω1, ϕ1, c1 are solutions for optimal wealth, hedge demand and the consumption-wealth ratio

in our original economy of interest p = 1.

We define functions Gn,Φn,Γn : I → R for n = 1, 2, 3, . . . by setting

Gn = χ ln ε+n∑k=1

gk,Φn =n∑k=1

ϕk and Γn = δψεη +n∑k=1

ck. (29)

Theorem 11 Assume a suitable 0 < p < 1 such that gp is an analytic function on the entire set I,for all |p−1| < p. Then we know that g1 = limn→∞Gn, ω1 =

λSγσS

+ϕ1, ϕ1 = limn→∞Φn, and c1 =

limn→∞ Γn.

This Theorem shows that the terms Gn,Φn,Γn provide approximations of the function g, the

hedge-demand ϕ and the consumption-wealth ratio c, respectively, in our original economy of

interest.

15

Proposition 12 For all k = 1, 2, . . ., the functions gk, ϕk and ck are separable in time and the

state-variable and the time dependence is of power k, i.e. there functions hg,k, hϕ,k, hc,k ∈ C1,2(I)such that gk(t, x) = hg,k(x) · (T − t)k, ϕk(t, x) = hϕ,k(x) · (T − t)k and ck(t, x) = hc,k(x) · (T − t)k.

Proposition 12 together with Theorem 11 tells us that we can interpret all summation terms

Gn,Φn,Γn for n = 1, 2, 3, . . . in these approximations: they collect all terms of order k = 1, 2, . . . , n

in the remaining lifetime T − t. For that reason we always refer to them as approximation terms of

order (up to) n. (While expansion terms of any order are separable in time and the state-variable,

this does not imply that hedge demand or the consumption-wealth ratio are separable in time and

the state-variable.)

Proposition 13 We denote β = max|β2|, |β2 + 1|, |2β3|, |ν + β3 − β1|, |2ν − 2β1|. For k = 1, . . .,

we define the set of base functions on I

Sk = (x, t) 7→ xi/2(T − t)k|(t, x) ∈ [0, T ]× [xl, xu];−kβ ≤ i ≤ kβ, i integer .

For all k = 1, 2, . . ., the functions gk and ck can be represented through the set of base functions

Sk; the function ϕk can be represented through the set of base functions Sk+2.

Proposition 12 provides a structural decomposition but remains short of a description through

basis functions. Proposition 13 specifies basis functions that represent the functions hg,k, hϕ,k, hc,k

(k = 1, 2, . . .). (Note that it does not ensure a sparse representation through basis functions; in

typical applications it is possible to use a much smaller set of base functions.) Therefore, one may

want to see our perturbation as a representation of the function g (and of hedge demand) in terms

of a (two-dimensional) series expansion using appropriate basis functions.

It is important to stress that our perturbation expansion relates the basis functions to the

powers in the functional form of drift and volatility functions; this can be seen clearly in the

proof of Proposition 13 and it will also become clear in the next two subsections. However, these

drift/volatility functions are power functions in the state-variable, where the power is a multiple

of 1/2 (here); this suggests an expansion in appropriate square root terms. In that regard, our

approach differs from other approachs, e.g., from Schwartz and Tebaldi (2006) and, on a more

theoretical level from Navasca (2002). For example, Navasca (2002) carries out such a power series

expansion of the coefficient functions in a Dynamic Programming Equation (DPE), plugs these in

the DPE and matches terms of similar power, in an extension of Al’Brekht (1961). In our setup,

Navasca (2002) would correspond to a power series expansion of the ξi (i = 1, 2, 3, 4) functions

around x = 0, giving a power series expansion of the solution around x = 0. However, we carry

out a series expansion in the additional perturbation parameters and end up with solutions where

the basis function are determined by that of the diffusion functions, i.e. here power function with

non-integer powers (potentially).

16

4.2 Iterative Steps for Asset Allocation Problems

Let us now present the iterative steps separately for asset allocation and consumption-savings

problems. We recall that the ξi functions are given in equations (27, 28); based on (25) we calculate

first for 0 ≤ t ≤ T, xl ≤ x ≤ xu:

g1(t, x) = −ξ1(x) · (T − t) = −(T − t) ·(r(γ − 1) +

γ − 1

2γσ2Sλ2x2ν−2β1

). (30)

Based on equation (25), higher order terms (k = 1, . . .) are determined iteratively through

gk+1 = −ξ2(x)∫ T

t

∂gk∂x

dt−k−1∑j=1

ξ3(x)

∫ T

t

∂gj∂x

∂gk−j∂x

dt− ξ4(x)

∫ T

t

∂2gk∂x2

dt. (31)

The procedure can be calculated out easily in computer algebra software. For illustration let us

present the calculation for k = 1, in order to determine g2. Based on equation (30) we find

∂g1∂x

= −(T − t)(γ − 1)λ2 (ν − β1)

γσ2Sx−2β1+2ν−1, (32)

∂2g1∂x2

= −(T − t)(γ − 1)λ2 (−2β1 + 2ν − 1) (ν − β1)

γσ2Sx−2β1+2ν−2, (33)

and calculate

g2(t, x) = −∫ T

tξ2(x)

∂g1∂x

+ ξ4(x)∂2g1∂x2

dt.

= (T − t)2(γ − 1)λ2 (ν − β1)

4γ2σ3Sx−3β1+2ν−2 (34)

·(γxβ1 σS

(2κ (x− x)xβ2+1 + (2β1 − 2ν + 1)x2β3 σ2X

)+ 2(γ − 1)λρσXx

β3+ν+1).

(There are no product terms based on∑k−1

j=1∂gj∂x

∂gk−j∂x for k = 1 (here), since the summation is

empty; product terms, however, will typically appear in all higher order terms.) In the interest of

saving space, we refrain from presenting terms of order higher than k = 1 for gk+1.

At the end of the previous subsection 4.1 we explained that solutions can be represented using a

set of appropriate basis functions and stressed that this differs from a simple power series expansion

(basis functions xj , j = 0, 1, . . .), in general. Equations (30, 34) allow us to study only the first

two expansion terms but they illustrate our point: they suggest that we need to consider terms xj

of powers j = 0, 2ν − 2β1 for the first order term and terms xj of powers j = −3β1 + β2 + 2ν −1,−3β1 + 2β3 + 2ν − 2,−3β1 + β3 + 3ν − 1. We recall that ν ≥ β1, β2, β3 are non-negative integer

multiples of 1/2, such that g2 may contain terms of half-power in x. Such terms would not show

up in a simple power series expansion.

Based on equation (22) together with the above representations g1, g2 we calculate the second

17

order approximation of hedge demand,

ΦAA2 =ρσXγσS

∂g1 + g2∂x

=ρσXγσS

xβ3−β1∂g1 + g2∂x

= −(T − t)(γ − 1)λ2ρσX

γ2σ3S(ν − β1)x

−3β1+β3+2ν−1

+(T − t)2(γ − 1)λ2ρσX

2γ3σ4S(ν − β1)x

−4β1+β3+2ν−3

·((γ − 1)λρ (−3β1 + β3 + 3ν − 1) σXx

β3+ν+1 + κγσS (x− (x− x)(−2β1 + β2 + 2ν))xβ1+β2+1

+(2β1 − 2ν + 1) (−β1 + β3 + ν − 1) γσS σ2Xx

β1+2β3). (35)

We do so for reference, only. Higher order approximation could be presented in closed-form, but to

save space we present only the second order approximation. This second order (as well as higher

order) approximations could be used for further structural analysis but we refrain from doing so.

4.3 Iterative Steps for Consumption-Savings Problems

For consumption-savings problems we have χ = 1; compared to the asset allocation problem of the

previous subsection this adds additional terms to equation (25). We calculate for 0 ≤ t ≤ T, xl ≤x ≤ xu:

g1(t, x) = −(ξ1(x) + δθ − δψ

ηεη)(T − t)

= −(T − t)

(r(γ − 1) +

γ − 1

2γσ2Sλ2x2ν−2β1 + δθ − δψ

ηεη). (36)

We compare this with g1 in the asset allocation problem, see equation (30) and note that constants

are added; therefore ∂g1∂x ,

∂2g1∂x2

are as in equations (32, 33). To determine g2 this amounts to adding

the time integration of δψεηg1, see equations (23, 25). Using the representation (34) for the asset

allocation problem, we find:

g2(t, x) = −∫ T

tξ2(x)

∂g1∂x

+ ξ4(x)∂2g1∂x2

dt− δψεη g1(x)dt.

= (T − t)2(γ − 1)λ2 (ν − β1)

4γ2σ3Sx−3β1+2ν−2 (37)

·(γxβ1 σS

(2κ (x− x)xβ2+1 + (2β1 − 2ν + 1)x2β3 σ2X

)+ 2(γ − 1)λρσXx

β3+ν+1)

+(T − t)2δψεη(r(γ − 1) +

γ − 1

2γσ2Sλ2x2ν−2β1 + δθ − δψ

ηεη).

(This differs from the g2 function in the asset allocation problem. see equation (34), only by the

term in the last line.) Based on equation (25), higher order terms (k = 1, . . .) are determined

18

iteratively

gk+1 = −ξ2(x)∫ T

t

∂gk∂x

dt−k−1∑j=1

ξ3(x)

∫ T

t

∂gj∂x

∂gk−j∂x

dt− ξ4(x)

∫ T

t

∂2gk∂x2

dt (38)

+δψεηk∑

n=0

ηn−1

n!

∑j1,...,jn≥1j1+...+jn=k

∫ T

t

n∏m=1

gjmdt.

This iterative procedure may look difficult to carry out; however, it only involves taking deriva-

tives and multiplication together with additive operations in a well-defined way; the different loops

(summations, products) can be implemented directly in symbolic algebra programs. In conse-

quence, this permits us to determine all expansion functions in closed-form in an iterative way.

(Compared to asset allocation problems, consumption-savings problems involve additional calcula-

tions in each iterative steps, see the second line in equation (38). Therefore, the implementation

of consumption-savings problems in symbolic algebra software is computationally more demanding

than asset allocation problems.)

We noted above the terms in g2 that differ in the asset allocation and the consumption-savings

problem. Using the second order approximation ΦAA2 the asset allocation, see equation (35), as well

as the derivative of g1 in equation (32) we find the second order approximation of hedge demand :

ΦCS2 = ΦAA2 − ρσXγσS

∂∫ Tt δψεηg1dt

∂x= ΦAA2 − ρσX

γσSδψεηxβ3−β1

∫ T

t

∂g1∂x

dt

= ΦAA2 − (T − t)2(γ − 1)λ2ρσX

2γ2σS3(ν − β1) δ

ψεηxβ3−3β1+2ν−1. (39)

Equations (22, 29) give the second order approximation of the consumption-wealth ratio:

Γ2 = δψεη(1 + 2ηg1 +

η2

2g21

).

We report here the second order approximations ΦCS2 ,Γ2 for illustration of our approach, only.

To save space we do not report higher order approximations. They do provide a closed-form

approximation for hedge-demand and the consumption-wealth ratio that could be used for further

structural analysis, but we refrain from doing so as it is beyond the scope of this paper.

5 Numerical Evaluation

This section discusses the numerical properties of our perturbation approach to asset allocation and

consumption-savings problems when the market price of risk is stochastic, see Kim and Omberg

(1996). Throughout this section we study the functional sepcification λS(t, x) = λx, σS = λ and

µX(t, x) = κ(x− x),σX(t, x) = σX .

19

5.1 General Issues

We use the parameter estimate of the market price of risk process given by Wachter (2002), based

on Campbell and Viceira (1999): λ = 0.0436, σX = 0.0189, κ = 0.0226, x = 0.0788. (These

are monthly values; annual time multiplied by 12 gives the number of months in computations.)

Finally, we adopt the correlation ρ = −0.93 that Barberis (2000) reports and choose an interest rate

r = 0.25%. Note that the myopic demand is given by 7.6453 times the market price of risk x for our

set of parameters. We choose the relative risk aversion parameter γ = 3; in consumption-savings

problems we set δ = 0.01, ε = 1 and study two cases ψ = 0.8, 1.2 for the elasticity of intertemporal

substitution. (These preference parameters are typical choices in financial applications.)

From a financial perspective, we are mostly interested in the situations where the market price of

risk for a time-horizon of one month (annualized) is within the interval [0.1, 0.3] (within the interval

[0.3461, 0.6928]); smaller/larger market prices of risk are not common in stock markets. This leads

us to set xl = 0.1;xu = 0.3. We calculate ||ξ1||[xl,xu] = 0.0035; ||ξ2||[xl,xu] = 0.001484; ||ξ3||[xl,xu] =0.00007562; ||ξ4||[xl,xu] = 0.0001786. For our choice of parameters, the conditions of Theorem 10

are fulfilled, i.e. our perturbation approximates the solution to the operator equation (20). By

Theorem 7 the solution to the operator equation (20) characterizes the optimal portfolio (and

consumption-savings decision) for state variables on the subset [xl, xu].

We look at relative errors in the logarithmic transformation function g and in hedge demand ϕ;

for this we define maximum relative errors of our approximations for g and ϕ, at order n:

Egn =

∣∣∣∣∣∣∣∣Gn − g

g

∣∣∣∣∣∣∣∣∞, and Eϕn =

∣∣∣∣∣∣∣∣Φn − ϕ

ϕ

∣∣∣∣∣∣∣∣∞, (40)

where the norm || · ||∞ describes the maximum on the compact set I, see subsection 3.4.

In the appendix we report the closed-form expression for the logarithmic transformation function

g and hedge demand in the asset allocation problem and use this as the true solution g (ϕ) in the

error calculation. To our knowledge, there is no known general closed-form expression in the

consumption-savings problem: notable exceptions are Wachter (2002) and Kraft et al. (2013). The

former studies the case of CRRA preference and correlation ρ = −1 (complete-markets case),

while the latter looks at a particular parametrization with ψ = 2 − γ + (1−γ)2γ ρ2. Both represent

the logarithmic transformation function g as a time-integral that needs to evaluated numerically.

Our interest focuses Kraft et al. (2013) but their integrand is an integral that needs to evaluated

numerically itself, see their appendix D; this created serious numerical issues. In our implementation

of the consumption-savings problem (below) we determine the true value for g and its derivative

(ϕ) through Mathematica’s NDSolve routine that solves numerically the PDE Lg = Gg for g; we

then use this to determine the relative error terms Egn, Eϕn for approximations at different orders n.

Although our focus is on portfolio selection, we recall that it is necessary to ensure convergence

for our approximation Gn to the function g, i.e. that our perturbation approach is valid. In addition

to our theoretical result in Theorem 10, we intend to cross-check heuristically (in applications)

whether gp is analytic on I = [0, T ] × [xl, xu] for p = 1. We recall that the (functional) series

20

limn→∞∑n

k=1 gk converges if limn→∞∑n

k=1 ||gk||∞ converges. To check heuristically convergence

of the latter, we define for all orders of our approximation n = 1, 2, . . . two terms Tn,Rn that look

at the convergence of our approximation of the g function:

Tn =||gn+1||∞||gn||∞

, and Rn = (||gn||∞)1n . (41)

Our definitions of Tn,Rn check the convergence n→ ∞ of∑n

k=1 ||gk||∞ in two ways: The first

(Tn) relates to the so-called ratio test, while the second (Rn) relates to the so-called root test. For

both tests, it is well known that the underlying series converges (diverges), if the limit superior

lim supn→∞

is smaller (larger) than one; either test is inconclusive, if the limit superior converges to one.

We know from there that our approximation for g = g1 converges if the limit superior for either

series (Tn or Rn) is smaller than one. (If this is strictly smaller than one, we could extend this to

show convergence for all p in a suitably small neighborhood of p = 1, i.e. that gp is locally analytic.

We refrain from doing so here.) Note that Tn,Rn link to convergence for our perturbation based

approximation of the logarithmic transformation function g, only; however, when convergence for

g holds, convergence of hedge demand follows.

To provide an alternative view on the performance of our perturbation6 we compare the expected

utility level derived from two portfolio selection strategies: one is our perturbation approximation

and the other the myopic strategy. To determine the expected utility we carry out a Monte-Carlo

simulation within Mathematica: for this we simulate the 10, 000 paths of the bivariate stock and

state variable process with time step 0.1 months (0.00833 years). In this simulation, we would have

to study the redefined the process; instead we adopt a practitioners approach: our perturbation

approach provides a functional description and we use that in our simulation; to counter problems

with unbounded paths of the state-variable, we impose an ad-hoc upper bound of 1 and a lower

bound of −0.5 on the consumption-wealth ratio.

5.2 Asset Allocation Problems

This subsection studies asset allocation problems, while the next one looks at consumption-savings

problems. Figure 1 presents the logarithmic value (to the base 10) of the maximum relative error:

Panel (a) looks at the error Egn in approximating the logarithmic transformation function g and

Panel (b) at the error Eϕn in approximating hedge demand ϕ. Both Panels study the terms at the

orders n = 1, 2, . . . , 10 and four different situations for the remaining time T − t = 6, 12, 18, 24

months (corresponding to 0.5, 1, 1.5, 2 years).

We see that both the logarithmic transformation function g and the hedge demand ϕ can be

calculated accurately using low order approximations. For example, to attain a maximum relative

error of less than 1%, i.e. log terms smaller than −2, a second-order approximation (n = 2) for the

6Den Haan and Marcet (1994) studied the accuracy of simulation methods that solve stochastic dynamic pro-gramming, Haugh et al. (2006) provide an upper bound on the utility loss associated with an approximate solution,and Jin and Judd (2002) study how well an approximation fits the HJB equation. We focus on the utility criterion,since it links directly to our objective.

21

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

à

à

à

à

à

à

à

à

à

à

ìì

ì

ì

ì

ìì

ì

ì

ì

òò

òò

ò

òò

ò

ò ò

1 2 3 4 5 6 7 8 9 10-14

-12

-10

-8

-6

-4

-2

0

Order n

log

max

rel.

erro

r

(a) Egn

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

à

àà

à

à

à

à

à

à

à

ì

ìì

ì

ìì

ì

ìì

ì

ò

òò

ò

òò

ò

òò

ò

1 2 3 4 5 6 7 8 9 10-14

-12

-10

-8

-6

-4

-2

0

Order n

log

max

rel.

erro

r

(b) Eϕn

Figure 1: Logarithm of the maximum relative error depending on the order of the approximationn; asset allocation problem; remaining time T − t is 6 months ( ), 12 months (), 18 months ()and 24 months (N).

g function approximation is fine when T − t = 0.5, 1, and a third-order approximation (n = 3) is

sufficient for all T − t values that we study here; for the hedge demand approximation, a second-

order approximation (n = 2) is sufficient for this level of relative accuracy, for all our choices of

the remaining lifetime. The relative accuracy gets better, as we increase the order, and the error

decreases faster the smaller T − t. Finally, we note that at order n = 10 our perturbation method

yields approximations that are accurate up to a relative error of roughly 10−6 (at worst) for all

remaining lifetimes T − t studied here.

While Theorem 10 proves convergence from a theoretical perspective, we want to take another

look at this heuristically. Panel (a) in Figure 2 presents our ratio criterion Tn and Panel (b) our

root criterion Rn, see equation (41). As in figure 1, both Panels study the terms at the orders

n = 1, 2, . . . , 10 and four different situations for the remaining time T − t = 6, 12, 18, 24 months

(corresponding to 0.5, 1, 1.5, 2 years).

Orders n = 3, 6, 9 show much larger values for Tn (Panel (a)) than the orders n; while the latter

appear somewhat bounded, the values for n = 3, 6, 9 appear to increase and might potentially lead

(in the limit n → ∞) to a supremum that is not strictly smaller than one; this suggests that the

ratio test is inconclusive. Fortunately, the root test criterion shows a different picture. For all four

cases of remaining time we see in Panel (b) that Rn decreases as the order n increases; moreover, it

appears to level off and that remaining level is larger for larger T−t. This suggests that convergenceholds but that its “speed” is worse, the larger T − t; this in line with our earlier observations based

on figure 1.

To provide an alternative view on the performance of our perturbation approach, we compare

the utility J(t,W0, x0) = E

[W 1−γT

1−γ

]that the agent derives from implementing our perturbation

approximation with the utility that she derives from the myopic strategy, a (potentially) suboptimal

portfolio selection strategy: Panel A in Table 1 looks at our perturbation approximation and Panel

22

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æ

æ

æ æ

æ

ææ

æ

æà

à

à

àà

à

à

à

à

à

ì

ì

ì

ìì

ì

ì

ì

ì

ì

ò

ò

ò

ò

ò

ò

ò

ò

ò

ò

1 2 3 4 5 6 7 8 9 10

0.2

0.4

0.6

0.8

1.

Order n

Rat

ioC

rite

rion

(a) Ratio Test Criterion Tn

æ

æ æ æ æ æ æ æ æ æ

à

à àà

àà

à à à à

ì

ì ìì

ìì

ì ìì

ì

ò

ò òò

òò

ò òò

ò

1 2 3 4 5 6 7 8 9 10

0.2

0.4

0.6

0.8

1.

Order n

Roo

tCri

teri

on

(b) Root Test Criterion Rn

Figure 2: Heuristic evaluation of convergence for the logarithmic transformation function g depend-ing on the order of the approximation; asset allocation problem; remaining time T − t is 6 months( ), 12 months (), 18 months () and 24 months (N).

Table 1: Expected utility J and certainty equivalents at various horizons T (expressed in months);asset allocation problem.

Panel A: Perturbation

Utility (×104) Certainty Equivalent

n T = 6 T = 12 T = 18 T = 24 T = 6 T = 12 T = 18 T = 24

1,. . . ,10 -0.4480 -0.4021 -0.3585 -0.3217 105.6 111.5 118.1 124.7

Panel B: Myopic

Utility (×104) Certainty Equivalent

T = 6 T = 12 T = 18 T = 24 T = 6 T = 12 T = 18 T = 24

-0.4480 -0.4022 -0.3593 -0.3235 105.6 111.5 118.0 124.3

B at the myopic strategy. (At four digits the results for the perturbation are the same at orders

n = 1, 2, . . . , 10.) Throughout we set initial valuesW0 = 100, x0 = 0.2. Table 1 also presents the so-

called certainty equivalents W0, defined as the initial wealth level that solvesW 1−γ

01−γ = J(t,W0, x0).

We recall that (for fixed horizon T ) higher utility as well as larger certainty equivalent are preferable.

We find for all horizons T that utility and certainty equivalents are very accurate right from the

start, in line with our earlier discussion of figure 1. At small horizons, with the shown four digit

results, there appears no difference between the myopic strategy and our perturbation approxima-

tion; this should not be surprising since the hedge demand should becomes more relevant at longer

time horizons; in fact table 1 confirms this and shows that our perturbation approach captures this.

Overall, utility derived and the associated certainty equivalents for our perturbation approach are

at least as good as that from the myopic strategy or higher.

23

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æ

æ

ææ æ æ æ æ æ

à

à

à

àà à à à à à

ì

ì

ì

ì

ìì ì ì ì ì

ò

ò

ò

ò

òò ò ò

òò

1 2 3 4 5 6 7 8 9 10-4

-3

-2

-1

0

Order n

log

max

rel.

erro

r

(a) Eg

æ

æ

æ

ææ æ æ æ æ æ

à

à

à

àà à à à à à

ì

ì

ì

ì

ìì ì ì ì ì

òò

ò

ò

òò ò

òò

ò

1 2 3 4 5 6 7 8 9 10-4

-3

-2

-1

0

Order n

log

max

rel.

erro

r

(b) Eϕ

Figure 3: Logarithm of the maximum relative error depending on the order of the approximationn; consumption-savings problem with ψ = 0.8; remaining time T − t is 6 months ( ), 12 months(), 18 months () and 24 months (N).

5.3 Consumption-savings Problems

While the previous subsection studied asset allocation problems, this one looks at consumption-

savings problems. Figures 3 and 4 present the logarithmic value (to the base 10) of the maximum

relative error when the elasticity of intertemporal substitution is ψ = 0.8 and ψ = 1.2, respectively.

Both figures are organized analogous to figure 1. Note that we have η = −0.1 and Theorem 10 is

applicable, if ψ = 0.8. With ψ = 1.2 we have η = 0.1 and the sufficient condition in Theorem 10 is

fulfilled, since δψεη exp(η)/η − δθ = 0.076 and ||ξ1||[xl,xu] = 0.0070.

Figures 3 and 4 show that our perturbation approach attains a good level of relative accuracy

(10%/1%, corresponding to log scale -1/-2) with a low order approximation. Also note that our

perturbation method provides better approximations for smaller lifetime T − t; relative accuracy is

better with ψ = 1.2 than with ψ = 0.8. We attribute this to the way we derived our series approx-

imation of cp = δψ exp(ηgp) in equation (22): we carried out a simple Taylor-series approximation

of the exponential function, see Proposition 9; it is well known that numerical issues arise with

negative values; note that ψ = 0.8 and ψ = 1.2 lead to different signs for the parameter η.

The maximum relative errors in approximating the g function and the hedge demand appear

to level off. We studied this puzzling feature and tweaked the Mathematica NDSolve routine in

various ways suggested by its documentation: in general, this affected the accuracy level at which

our perturbation approach levels off but did not resolve it entirely. Kraft et al. (2013) describe

a so-called closed-form expression in some special cases, but their formula involves a numerical

integration of a function that needs to be integrated numerically; it suffers from considerable

numerical issues7, but does not provide additional insights, so that we do not report it here.

Figure 5 studies convergence heuristically using the root test criterion: Panel (a) looks at

7Strikingly, Kraft et al. (2013) also provide a similar closed-form expression in the Heston stochastic volatilitymodel. In a follow-up paper Kraft et al. (2014) derive a numerical method to portfolio selection for that model;however, they do not use the closed-form expression suggested in Kraft et al. (2013), neither.

24

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æ

æ

æ æ æ æ æ æ æ

à

à

à

à à à à à à à

ì

ì

ì

ì ì ì ì ì ì ì

ò

ò

ò

ò ò ò ò ò ò ò

1 2 3 4 5 6 7 8 9 10

-5

-4

-3

-2

-1

0

Order n

log

max

rel.

erro

r

(a) Eg

æ

æ

æ

æ æ æ æ æ æ æ

à

à

à

à à à à à à à

ì

ì

ì

ì ì ì ì ì ì ì

ò

ò

ò

ò ò ò ò ò ò ò

1 2 3 4 5 6 7 8 9 10

-5

-4

-3

-2

-1

0

Order n

log

max

rel.

erro

r

(b) Eϕ

Figure 4: Logarithm of the maximum relative error depending on the order of the approximationn; consumption-savings problem with ψ = 1.2; remaining time T − t is 6 months ( ), 12 months(), 18 months () and 24 months (N).

ψ = 0.8, while Panel (b) studies ψ = 1.2. (We do not study the ratio test criterion since it turned

out “inconclusive,” again.) This discussion can only provide results that are suggestive and, clearly,

we cannot prove convergence. However, figure 5 suggests for both cases that the limit superior is

less than 1 and that convergence holds. (Also, we see much better results for ψ = 1.2 in Panel (b)

than for ψ = 0.8 in Panel (a); this is in line with our observation above.)

We noted earlier that Theorem 10 applies, i.e. convergence is assured; our heuristic cross-

check using the root criterion confirms this. Therefore, we attribute it to numerical issues of the

Mathematica NDSolve routine, that errors level off in Figures 3 and 4. Further analysis of this

leveling off is recommended; in addition better approximations (other functional series expansion)

for cp should be studied when η < 0 (ψ < 1).

In the remainder of this subsection, we discuss the performance of our perturbation approach in

terms of utility and certainty equivalents for an investor with ψ = 1/γ (CRRA preferences)8: the

normalized integrator boils down to e−δ(T−s)C1−γs1−γ such that the indirect utility can be implemented

through a Monte-Carlo simulation of stock and state variable paths. Analogous to our analysis in

the previous subsection, we carry out a Monte-Carlo simulation with starting valuesW0 = 100, x0 =

0.2.

Our perturbation approach leads to an approximation of g by Gn at various orders n =

1, 2, . . . , 10 and this provides a functional description of portfolio demand and consumption at all

times and across all states. Our goal is to compare this with the myopic strategy; the latter is char-

acterized by g = 0, s.t. hedge demand ϕ = 0 and consumption-wealth ratio c = δψ exp(ηg) = δ1/γ .

To determine the certainty equivalent, we compare this with the case where the investor would

invest all her wealth in the riskless asset and consume a constant fraction c = δ1/γ of it at all times.

8For investors with ψ = 1/γ (Epstein-Zin preferences), we would have to determine the indirect utility J inequation (6) for a given portfolio and consumption strategy. However, this involves an expectation over a processand the process depends on knowing the function J , itself; this is not directly amenable to a Monte-Carlo simulation.

25

æ

æ æ æ æ

æ ææ æ

à

àà

àà

à à

à à

ìì

ìì

ìì

ì ì

òò

ò ò

1 2 3 4 5 6 7 8 9 10

0.2

0.4

0.6

0.8

1.

Order n

Roo

tCri

teri

on

(a) ψ = 0.8

æ

ææ

ææ æ æ æ æ æ

à

àà

à à à à à à

ì

ì

ì

ì ì ì ì ì ì

ò

ò

ò

ò ò ò ò ò ò

1 2 3 4 5 6 7 8 9 10

0.2

0.4

0.6

0.8

1.

Order n

Roo

tCri

teri

on

(b) ψ = 1.2

Figure 5: Heuristic evaluation of convergence using the root test criterion depending on the orderof the approximation; consumption-savings problem; remaining time T − t is 6 months ( ), 12months (), 18 months () and 24 months (N).

A straightforward calculation shows that the investor would derive utility

J(W0) =W 1−γ

0

1− γ

c1−γ

−δ + (1− γ)(r − c)

(e(−δ+(1−γ)(r−c))T − 1

)+ εe(1−γ)(r−c)T

.

For a given utility level J , the certainty equivalent is then defined as the wealth level W0 with the

property that J(W0) = J .

Table 2 presents the indirect utility and certainty equivalent when the investor follows the

portfolio selection (and consumption-savings decision) according to our perturbation approximation

at various orders n = 1, . . . , 10 (Panel A), and when she follows the myopic strategy (Panel B). For

low orders n the perturbation is (initially) inferior to the myopic, but it picks up quickly and yields

much higher levels of utility and of the certainty equivalent. (Comparing this with the previous

subsection, this suggests also that to dominate the myopic strategy, a higher order n is required for

consumption savings problems than for asset allocation problems.) We observe that for n = 10, for

all T studied here but with the exception of T = 6, the levels of utility and certainty equivalent are

much higher than those for the myopic strategy. (We attribute this exception to our ad-hoc bounds:

taking them out leads to out-performance for T = 6 of our perturbation approach compared to the

myopic strategy but to under-performance for all other time-horizons; as we are more interested

in longer time-horizons, we present results for this case, only.) Overall, our analysis shows that

our perturbation approach achieves significant improvements beyond the myopic strategy, both in

terms of utility and certainty equivalent.

6 Conclusion

This paper studied asset allocation and consumption-savings problems for models with stochastic

investment opportunities, including stochastic market prices of risk and stochastic volatility. Pref-

26

Table 2: Expected utility J and certainty equivalents at various horizons T (expressed in months);consumption-savings problem.

Panel A: Perturbation

Utility Certainty Equivalent

n T = 6 T = 12 T = 18 T = 24 T = 6 T = 12 T = 18 T = 24

1 -0.03128 -0.6448 -12.69 -251.1 97.1 77.4 61.0 47.82 -0.03121 -0.6435 -12.67 -250.6 97.2 77.5 61.1 47.83 -0.03121 -0.6435 -12.66 -250.6 97.2 77.5 61.1 47.84 -0.03121 -0.6435 -12.66 -250.6 97.2 77.5 61.1 47.85 -0.03121 -0.6435 -7.73 -45.1 97.2 77.5 78.2 112.76 -0.03121 -0.2869 -1.69 -9.8 97.2 116.1 167.4 241.87 -0.03121 -0.2237 -1.32 -7.7 97.2 131.5 189.2 273.38 -0.03015 -0.2063 -1.22 -7.1 98.9 136.9 196.9 284.39 -0.02919 -0.2005 -1.19 -6.9 100.5 138.8 199.6 288.210 -0.02923 -0.2007 -1.19 -6.9 100.4 138.8 199.5 288.0

Panel B: Myopic

Utility Certainty Equivalent

T = 6 T = 12 T = 18 T = 24 T = 6 T = 12 T = 18 T = 24

-0.02904 -0.3817 -4.67 -57.1 100.7 100.6 100.6 100.2

erences are specified as power utility derived from terminal wealth as well as consumption-savings

problems with recursive utility Epstein-Zin preferences. We introduced an approach based on per-

turbing the coefficients of the underlying stochastic dynamics, in a way that is flexible enough to

handle these setups in a general way. The perturbation approach reduces the problem to a sequence

of PDEs to be solved iteratively; this provides in an iterative way, at any desired accuracy, a series

of closed-form approximations that converge to the optimal hedge demand and to the consumption-

wealth ratio. In an error analysis we found that lower order (closed-form) approximations yield

accurate descriptions of the consumption-wealth ratio and, in particular, of hedge demand. This

provides an avenue for future analysis of the structural properties of hedge demand and portfolio se-

lection. A comparison of utility and certainty equivalents confirms that our approximative portfolio

selection strategies yield significant improvements in terms of utility and certainty equivalents.

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Appendix

A Supplementary Material

A.1 General Properties of Solutions to Special Cases

To illustrate, how our approach can be used to study general properties of solutions, we revisit

special cases studied in the literature. For our first analysis, we note that the literature does not

agree on the choice of the parameter ν. We want to analyze the choice ν = β1; then we have that

the function ξ1, defined in equation (27) is a constant, i.e. it does not dependent on time or the state

variable. We then note that g1(t, x) = −ξ1(x) · (T − t) and that its dependence on the state variable

disappears. Similarly, we note also from our representation that the dependence of g2 on the state

variable disappears.) This means that their derivatives w.r.t. x vanish and it is straightforward

to check iteratively based on equation (25), together with the definition of Gk that the dependence

of gk on the state variable disappears. Note that the functions ξi (i = 1, 2, 3, 4) are the same

in asset allocation and consumption-savings problems as long as the underlying process structure

is identical; this holds both for asset allocation and consumption-savings decisions, although the

resulting constant may differ. Overall we find:

Proposition A-1 Assume ν = β1 and 0 < p < 1 such that gp is analytic for all |p− 1| < p. Then

for all k = 1, 2, . . .: gk = (T − t)khg,k for a suitable constant hg,k and ϕk = 0.

The option pricing literature usually adopts ν = β1 to carry out a probability measure trans-

formation via the Girsanov theorem. Within our radius of convergence, this Proposition shows

that this dependence vanishes in the g function and so that the hedge demand vanishes. From a

financial perspective, the condition ν = β1 means that the market price of risk is constant; our

result matches the insight in Kraft et al. (2013). This suggests that we cannot have ν = β1 for

interesting applications. In fact, portfolio applications of the Heston model (β1 = 1/2) set ν = 1.

For our second analysis, we look at Liu (2007); he showed in a multivariate framework that

the logarithmic transformation g is a quadratic polynomial when the interest rate r, the drift µX

and squared volatility function σ2X of the state process as well as (λ/σS)2 (in our notation) are all

quadratic polynomials of the state variable, see his equations (9, 10, 14). His specification does not

31

fit assumption 1, but fits into the analysis of section 3, subject to suitable technical restrictions.

Such an analysis would then find that the functions ξi (i = 1, 2, 3, 4), defined in equations (17, 18)

are quadratic polynomials in the state variable. Analogous to the inductive proof of this proposition,

it could then be shown for our univariate framework with constant interest rates that all gk terms

are quadratic polynomials; within our radius of convergence this would imply that the g function is

a quadratic polynomial, in line with Propositions 1 and 2 in Liu (2007) applied to asset allocation

problems (α = 0).

A.2 Closed-form Expressions for Stochastic Market Price of Risk

Kim and Omberg (1996) present the closed-form expression for the function g that can be used to

determine hedge demand ϕ. (They reduce the HJB equation to three PDEs of Riccati type and

show there are four types of solutions depending on q, but only the so-called “normal” solution for

q > 0 is relevant here.) For completeness, we report this here:

g(t, x) = h0(T − t) + h1(T − t)x+ h2(T − t)x2

2, (A-1)

such that ϕ(t, x) =ρσX(x)

γσS(x)

∂g

∂x=ρσXγλ

∂g

∂x=ρσXγλ

(h1(T − t) + h2(T − t)x) , (A-2)

where

k0 = −γ − 1

γ; k1 = −2

(γ − 1

γρσX + κ

); k2 = −

γ(ρ2 − 1

)− ρ2

γσ2X ;

k3 = κx; q = k21 − 4k0k2,

h2(t) =2k0(1− exp(−√

qt))

2√q − (k1 +

√q)(1− exp(−√

q t)),

h1(t) =4k0k3

(1− exp(−√

q t/2))2

√q(2√q − (k1 +

√q)(1− exp(−√

q t))) ,

h0(t) =

∫ t

0κxh1(t)−

γ(ρ2 − 1

)− ρ2

2γσ2Xh

21(t) +

σ2X2h2(t)− (γ − 1)rdt.

A.3 Properties of Implementation for Stochastic Market Price of Risk

Finally, let us derive interesting general properties of our implementation for stochastic market

prices of risk. In the asset allocation problem we recall that the ξi functions have been defined in

equations (17, 18); we then note that the function ξ1 is a quadratic polynomial in x, ξ2 is a linear

functions of x and ξ3, ξ4 are constants. Our general iterative procedure presented in the previous

subsection 4.2 then gives the functional series terms. The particular form of the ξi functions greatly

simplifies the implementation of the previous subsection and the reader can easily infer these from

equations (32, 34) by setting the model parameters accordingly. In the interest of saving space we

do not write out these here.

32

Proposition A-2 Consider an asset allocation problem with stochastic market price of risk in

the specification of Kim and Omberg (1996). Assume 0 < p < 1 such that gp is analytic for all

|p− 1| < p. Then, for all k = 1, 2, . . ., gk = (T − t)khg,k, where hg,k is a quadratic polynomial in x.

Proof of Proposition A-2. To see this, we note that ξ1(x) = r(γ−1)+ γ−12γ x

2, which tells us

that g1 is a quadratic polynomial in x; this shows that a quadratic polynomial suffices to describe

the functional dependence of g1 on the state variable. Taking first (second) order derivatives means

that their functional dependence is linear (constant) and this shows that a quadratic polynomial

suffices to describe the functional dependence of g2 on the state variable. This observation carries

through our (iterative) procedure in equation (31), since ξ2 is a linear function in x and ξ3, ξ4 are

constants. This shows that quadratic polynomials in x suffice to describe the functional dependence

on x in all gk terms. Overall, this shows (within the radius of convergence) that g = g1 itself is a

quadratic polynomial in x, i.e. that hedge demand is a linear function in the state variable x.

This proposition tells us, within the radius of convergence, that base functions are constants,

proportional and quadratic functions in x. (This is a stronger result than that of Proposition

13.) Note also that our observation matches the functional form in Kim and Omberg (1996); they

find that the function g is a quadratic function in x, with time-dependent coefficients. (The ξi

functions in the consumption-savings problem are unchanged to the asset allocation problem, but

our iterative procedure does not allow us to conclude that the consumption-savings problem can

be represented through functions that are quadratic polynomials in x.)

B Proofs

Proof of Propositions 3 and 6. Proposition 3 is a special case (p = 1) of proposition 6

and so we prove only the latter. Since the original drift and volatility functions λS , σS , µX , σX

are infinitely differentiable, the perturbed drift and volatility functions λS,p, σS,p, µX,p, σX,p are

infinitely differentiable; in particular growth and Lipschitz conditions are fulfilled. This ensures

existence and uniqueness of a continuous, strong solution to the asset dynamics (8, 9), see, e.g.

Theorem (6.2.2) in chapter 6 of Arnold (1992) or Theorem (2.4) in chapter IX of Revuz and Yor

(1999).

Next we prove that the process Xp stays within Ξ. We start showing that the lower bound 0 is

not attained. First, we note that in under assumption 5, µX,p, σ2X,p both scale proportionally in p.

Thus, the scale density sL defined for our original model is also the scale density in the perturbed

model (to be defined analogously). Under the condition in equation (4) we apply Lemma 6.3 (a), p.

231 in Karlin and Taylor (1981) and conclude that the lower bound xL is not attained. Analogously,

it can be shown that the upper bound xU is not attained.

Proof of Theorem 7. We recall from the literature, see, e.g. Pennacchi (2008) and Kraft

33

et al. (2013), that our goal is to solve the quasi-linear partial differential equation

0 = maxωp,Cp

∂Jp∂t

+ χpf(Cp, Jp) + ((ωpλS,p + rp)w − χpCp)∂Jp∂w

+ µX,p∂Jp∂x

+1

2ω2pw

2σ2S,p∂2Jp∂w2

+1

2σ2X,p

∂2Jp∂x2

+ ρwωpσS,pσX,p∂2Jp∂w∂x

, (B-1)

with boundary condition

Jp(T,w, ·) = exp(χ ln ε)w1−γ

1− γ. (B-2)

Plugging the conjecture (13) into the PDE (B-1) yields

0 = maxωp,Cp

∂gp∂t

+ (ωpλS,p + rp)(1− γ) + µX,p∂gp∂x

− γ(1− γ)

2ω2pσ

2S,p (B-3)

+1

2σ2X,p

(∂2gp∂x2

+

(∂g

∂x

)2)

+ ρωσSσX(1− γ)∂g

∂x+ χ

(f(C, J)

J− C

w(1− γ)

).

The first-order condition for the optimal portfolio weight ωp then gives the optimal wealth weights

in equation (16).

In the remainder of this proof let us study the consumption-savings problem; we calculate

f(Cp, Jp)

Jp= δθ

((CpWp

)1− 1ψ

exp(−gpθ

)− 1

). (B-4)

The first-order condition for consumption Cp gives

cp = δψ exp (ηgp) for ψ = 1. (B-5)

Plugging first the optimal consumption-wealth ratio into equation (B-4) and then plugging these

results, together with the above description of the optimal wealth weights into the simplified HJB

equation (B-3) leads to the stated (non-linear) partial differential equation.

Since gp is a continuous function on a bounded interval, the drift and volatility of the wealth

process (10) fulfill growth and Lipschitz conditions. This ensures existence and uniqueness of a

continuous, strong solution to the asset dynamics (8, 9), see, e.g. Theorem (6.2.2) in chapter 6 of

Arnold (1992) or Theorem (2.4) in chapter IX of Revuz and Yor (1999). In addition, the assumption

cp = δψ exp(ηgp) < 1 ensures that Cp,t = cp,tWp,t > 0. Using Lemma 6.3 (a), p. 231 in Karlin and

Taylor (1981) we see that Wp,t > 0 at all times 0 ≤ t ≤ T . This then implies also that Cp,t > 0 at

all times 0 ≤ t ≤ T . Therefore, the strategy is admissible, i.e. (ωp, Cp) ∈ Ap.

To ensure that the solution to the HJB equation (B-1) with boundary condition (B-2) is the

solution to the agent’s utility maximization problem (11), we apply the verification Theorems 3.1

and 3.2 of Kraft et al. (2013). For this it remains to check that their martingality condition holds;

34

it is sufficient to show the integrability condition

E

[∫ T

0

∣∣∣∣∣∣∣∣∂Jp∂w(t,Wp,t) · (rWp,t + ωtλS(t,Xp,t)Wp,t − χCp,t)

∣∣∣∣∣∣∣∣2 dt]<∞

for all weight processes (ωp,t)0≤t≤T and processes (χCp,t)0≤t≤T defined on I. (The process Wp is

the associated wealth process, defined in equation (10).)

To prove this, we note first that there exists a constant Kg,p s.t. gp,∂gp∂t ,

∂gp∂2 ,

∂2gp∂x2

and cp =

δψ exp(ηgp) are all uniformly bounded by Kg,p on the compact set I ⊂ R2, since gp ∈ C1,2(I). We

conclude from this, together with assumption 2 that weight ωp is bounded on I. Also, we find

(based on assumption 2) that

rWp,t + ωtλS(t,Xp,t)Wp,t − χCp,tWp,t

= r + ωsλS(t,Xp,t)− χcp,t (B-6)

is bounded on I. We denote by Kg,p the maximum of this bound and of Kg,p. This implies that

the wealth process is a martingale and that its expectation grows with the rate in equation (B-6).

Since the growth rate is bounded by Kg,p, we have that E[W ζp,t] < max1, exp(ζKg,pt) for all

ζ ∈ R, 0 ≤ t ≤ T .

Next, we note that our conjecture (13) implies∂Jp∂w = w−γ exp(gp). Then we have

E

[∫ T

0

∣∣∣∣∣∣∣∣∂Jp∂w(t,Wp,t) · (rWp,t + ωtλS(t,Xp,t)Wp,t − χCp,t)

∣∣∣∣∣∣∣∣2 dt]

= E

[∫ T

0

∣∣∣W 1−γp,t · (r + ωtλS(t,Xp,t)− χcp,t)

∣∣∣2 dt] ≤ K2g,pE

[∫ T

0W 2−2γp,t dt

]<∞.

This ends the proof.

Proof of Theorem 8. For f ∈ C1,2(I) we define f = f + χ ln ε. We have ∂f∂t = ∂f

∂t and note

that χεχη = χεη to find

ξ1 + ξ2∂f

∂x+ ξ3

(∂f

∂x

)2

+ ξ4∂2f

∂x2+ χ

(−δ

ψ

ηexp(ηf) + δθ

)= ξ1 + ξ2

∂f

∂x+ ξ3

(∂f

∂x

)2

+ ξ4∂2f

∂x2+ χ

(−δ

ψ

ηεη exp(ηf) + δθ

)Note that the transformation f = f + χ ln ε replaces the original boundary condition associated to

the PDE (14) by a zero boundary condition. Thus, a solution gp to the PDE (14) defines a solution

gp to the problem (20), and vice versa.

Proof of Proposition 9. According to Theorem 7, we have in consumption-savings problems

(χ = 1) that

cp = δψ exp(ηgp) = δψ exp(η ln ε) exp

(η

∞∑k=1

pkgk

)= δψεη exp

(η

∞∑k=1

pkgk

).

35

We note that ( ∞∑k=1

pkgk

)n=

∞∑i=n

pi∑

j1,...,jn≥1j1+...+jn=i

n∏m=1

gjm .

Using the series expansion of the exponential function we then find

cp = δψεη + δψεη limn→∞

n∑n=1

ηn

n!

∞∑i=n

pi∑

j1,...,jn≥1j1+...+jn=i

n∏m=1

gjm .

Fix a k = 1, 2, . . .. To collect terms of order k in p, we look for terms with i = k in the inner

summation∑∞

i=n(. . .) but this is only possible for all n = 1, 2, . . . , k in the outer summation

limn→∞∑n

n=1(. . .). This observation allows us to rewrite the terms of order k in p as

pkδψεηk∑

n=1

ηn

n!

∑j1,...,jn≥1j1+...+jn=k

n∏m=1

gjm = 0.

Lemma B-1 Assume the conditions of Theorem 10 hold. For all real numbers 0 ≤ ζ1 < ||ξ1||[xl,xu],and 0 ≤ ζ2 we define a function Fζ1,ζ2 : R2 7→ R, by setting for (p, φ) ∈ R2

Fζ1,ζ2(p, φ) =1

T(F1,p,ζ1,ζ2(φ)− F2,p(φ)) , where (B-7)

F1,p;ζ1,ζ2(φ) = −Tφ+ p(2||ξ1||[xl,xu] − ζ1 +

(||ξ2||[xl,xu] + ζ2 + ||ξ4||[xl,xu]

)φ+ ||ξ3||[xl,xu]φ

2),

F2,p(φ) = −χp(δψ

|η|εη exp(ηφ) + δθ

).

Then, for sufficiently large T , numbers 0 < ζ1 < ||ξ1||, 0 < ζ2, 0 < p < 1 and an analytic function

φ0,ζ1,ζ2 : (1− p, 1 + p) → R exist, such that Fζ1,ζ2(p, φ0,ζ1,ζ2(p)) = 0 for all |p− 1| < p.

Proof of Lemma B-1. We define for all 0 ≤ ζ1 < ||ξ1||[xl,xu] and 0 ≤ ζ2:

φ∗ζ1,ζ2 = −

||ξ2||[xl,xu] + ζ2 + ||ξ4||[xl,xu] − T

2||ξ3||[xl,xu], and

φ±,ζ1,ζ2 = φ∗ζ1,ζ2 ±

√(φ∗

ζ1,ζ2)2 − 2||ξ1||[xl,xu] + ζ1 .

Note that limT→∞ φ∗ζ1,ζ2

= ∞ and limT→∞ Fζ1,ζ2(p, φ) = −∞; thus, for sufficiently large T ,

the term in the square root is strictly positive for all 0 ≤ ζ1 ≤ ||ξ1||[xl,xu] and 0 ≤ ζ2 ≤ 1 and the

term φ±,ζ1,ζ2 is a well-defined real number. Then, the function F1,p=1;ζ1,ζ2 is a convex quadratic

polynomial with minimum at φ∗ζ1,ζ2

and zeros at φ±,ζ1,ζ2 .

The remainder of this proof proceeds through two steps. In our first step, we prove that for all

0 ≤ ζ1 ≤ ||ξ1||[xl,xu], 0 ≤ ζ2, a number φ0,ζ1,ζ2 exists that solves Fζ1,ζ2(1, φ0,ζ1,ζ2) = 0. Let us fix

36

numbers 0 ≤ ζ1 ≤ ||ξ1||[xl,xu], 0 ≤ ζ2 and prove this separately for both types of portfolio selection

problems. First, we study asset allocation problems (χ = 0); for this, 0 = Fζ1,ζ2(1, φ) = F1,p;ζ1,ζ2(φ)

is a quadratic equation in φ. This equation has the solutions φ±,ζ1,ζ2 defined above, which are real

numbers for sufficiently large T .

Next, we study consumption-savings problems (χ = 1); then finding a zero φ0,ζ1,ζ2 of Fζ1,ζ2 at

p = 1 is equivalent to finding φ0,ζ1,ζ2 that solves F1,p=1;ζ1,ζ2(φ0,ζ1,ζ2) = F2,p=1(φ0,ζ1,ζ2) (a fixed-point

problem). We study separately the cases η < 0 and η > 0. (The case η = 0 is impossible since

η = −ψ/θ and we assume throughout the paper 1 = ψ > 0.)

First, we look at consumption-savings problems with η > 0. The assumption in Theorem

10 then implies F2,p=1(φ = 0) = − δψ

η εη exp(η) − δθ > 2||ξ1||[xl,xu] ≥ F1,p=1;ζ1,ζ2(φ = 0) for all

0 ≤ ζ1 ≤ ||ξ1||[xl,xu] and 0 ≤ ζ2. Furthermore, we note that θ = −ψ/η < 0 and so F2;p=1 is a strictly

monotonically decreasing function on the positive real line with limφ→−∞ F2;p=1(φ) = −δθ > 0 and

limφ→+∞ F2;p=1(φ) = −∞. Since limφ→∞ F1,p=1;ζ1,ζ2 = ∞ and limφ→∞ F2;p=1 = −∞ this implies

the existence of a real number φ0,ζ1,ζ2 that solves F1,p=1;ζ1,ζ2(φ0,ζ1,ζ2) = F2,p=1(φ0,ζ1,ζ2).

For consumption-savings problems with η < 0, we note that θ = −ψ/η > 0 and so F2;p=1 is a

strictly monotonically increasing function on the positive real line with limφ→−∞ F2;p=1(φ) = −∞and limφ→+∞ F2;p=1(φ) = −δθ < 0. Recall that φ∗

ζ1,ζ2describes the minimum of the function

F1,p=1;ζ1,ζ2 ; then note that F1,p=1;ζ1,ζ2(φ∗ζ1,ζ2

) = 2||ξ1||[xl,xu]−ζ1−(φ∗ζ1,ζ2

)2 and that this tends to −∞as T tends to ∞. Thus, for sufficiently large T , this is smaller than F2,p=1. (For this, it is important

to note that the term F2,p=1 does not depend on T .) We then note that limφ→∞ F1,p=1;ζ1,ζ2 = ∞and limφ→∞ F2;p=1(φ) = −δθ < 0; this implies the existence of a real number φ0,ζ1,ζ2 that solves

F1,p=1;ζ1,ζ2(φ0,ζ1,ζ2) = F2,p=1(φ0,ζ1,ζ2).

It remains to show in our second and final step that suitable numbers ζ1, ζ2 and an analytic

function p 7→ φ0,ζ1,ζ2(p) (in a suitable neighborhood of p = 1) exist. To show this we apply the

(classical) implicit function theorem; for this we need to find a suitable ζ1, ζ2 within the permitted

range s.t.∂Fζ1,ζ2∂φ (1, φ0,ζ1,ζ2) = 0. We start with ζ1 = ||ξ1||[xl,xu]/2, ζ2 = 0. From our first step, we

know that Fζ1,ζ2(1, φ0,ζ1,ζ2) = 0. Also, we find

∂Fζ1,ζ2∂φ

(1, φ) = −1 +||ξ2||[xl,xu] + ζ2 + 2||ξ3||[xl,xu]φ+ ||ξ4||[xl,xu] + χδψεη exp(ηφ)

T.

If this is not equal to zero at φ = φ0,ζ1,ζ2 , then an application of the implicit function theorem

on Fζ1,ζ2 shows that an analytic function φ0,ζ1,ζ2(p) exists on a suitable neighborhood of p = 1.

However, if this is equal to zero at φ = φ0,ζ1,ζ2 , i.e. if∂Fζ1,ζ2∂φ (1, φ0,ζ1,ζ2) = 0, then we adjust ζ1, ζ2

as follows: We define a parameter ϵ =||ξ1||[xl,xu]2φ0,ζ1,ζ2

> 0 and set ζ1 = ϵφ0,ζ1,ζ2 =||ξ1||[xl,xu]

2 , ζ2 = ϵ.

(Note that 0 ≤ ζ1 ≤ ||ξ1||[xl,xu] and 0 ≤ ζ2.) The definition of Fζ1,ζ2 in equation (B-7) shows that

Fζ1,ζ2(1, φ0,ζ1,ζ2) = 0. Furthermore, we note that

∂Fζ1,ζ2∂φ

(1, φ0,ζ1,ζ2) =∂Fζ1,ζ2∂φ

(1, φ0,ζ1,ζ2) +ϵ

T=

ϵ

T> 0.

37

Now, an application of the implicit function theorem on Fζ1,ζ2 shows that an analytic function

φ0,ζ1,ζ2(p) exists on a suitable neighborhood of p = 1. This ends the proof.

Proof of Theorem 10. Our goal is to use the method of majorants to prove convergence; to

define a suitable majorant equation, we define operators G 7→ C(I) by setting for f, f ∈ G:

M1f = ξ2∂f

∂x+ ξ4

∂2f

∂x2, M2(f, f) = ξ3

∂f

∂x

∂f

∂x, Mexpf = −χδ

ψ

ηεη exp(ηf).

We equip the space C(I) with the maximum norm || · ||∞, defined at the beginning of subsection

3.4. It is well known that differentiation operators G → C(I) of first and of second order in x have

norm 1. Therefore, for f, f ∈ G with ||f ||G = ||f ||G = 1, we know that

||M1f ||∞ ≤ ||ξ2||[xl,xu] ·∣∣∣∣∣∣∣∣∂f∂x

∣∣∣∣∣∣∣∣∞

+ ||ξ4||[xl,xu] ·∣∣∣∣∣∣∣∣∂2f∂x2

∣∣∣∣∣∣∣∣∞

≤ ||ξ2||[xl,xu] + ||ξ4||[xl,xu],

||M2(f, f)||∞ ≤ ||ξ3||[xl,xu] ·∣∣∣∣∣∣∣∣∂f∂x

∣∣∣∣∣∣∣∣∞

∣∣∣∣∣∣∣∣∣∣∂f∂x

∣∣∣∣∣∣∣∣∣∣∞

≤ ||ξ3||[xl,xu].

In addition, we know for f ∈ G that ||Mexpf ||∞ = χ δψ

|η|εη exp(η||f ||G). Set G = f ∈ C(I)|f(T, ·) =

0 and equip it with the maximum norm || · ||∞, defined at the beginning of subsection 3.4. For

f ∈ G we have ∣∣∣∣∫ t

Tf(s, x)ds

∣∣∣∣ ≤ ∫ t

T|f(s, x)|ds ≤

∫ t

T||f ||∞ds ≤ T ||f ||∞,

which shows that ||L−1|| = T .

We rewrite the original problem in the form

gp = pL−1

(ξ1 +M1gp +M2g

2p + χ

(−δ

ψ

ηεη exp(ηgp) + δθ

)).

Then, we proceed analogous subsection 8.3 in Zeidler (1993), see also subsection 3.3 in Berger

(1977), and consider the majorant equation

φ = pT

2||ξ1||[xl,xu] − ζ1 +

(||ξ2||[xl,xu] + ζ2 + ||ξ4||[xl,xu]

)φ+ ||ξ3||[xl,xu]φ

2

+χ

(δψ

|η|εη exp(ηφ) + δθ

),

for 0 ≤ ζ1 ≤ ||ξ1||Ξ, 0 ≤ ζ2. This is a majorant since 2||ξ1||Ξ − ζ1 ≥ ||ξ1||Ξ and ||ξ2||Ξ + ζ2 ≥ ||ξ2||Ξ.Solving this equation for φ is equivalent to solving Fζ1,ζ2(p, φ) = 0 for 0 ≤ ζ1 ≤ ||ξ1||Ξ, 0 ≤ ζ2,

defined in equation (B-7) of Lemma B-1. The structure of our problem implies that when a T

exists for which the statement holds with T = T , then it also holds for all T < T , since we can

restrict ourselves to smaller lifetimes. Therefore, w.l.o.g. we assume in the remainder of this proof

that T is sufficiently large to apply Lemma B-1. An application of Lemma B-1 then proves that a

38

suitable 0 < p < 1 exists, such that gp is analytic on I for all |p− 1| < p.

Equation (25) defines a series of functions that can be determined iteratively; thus, we prove

iteratively, that gk ∈ Ck,∞(I) for all k = 1, 2, . . .: To start, we note that, by assumption 2, the

function ξ1 is infinitely differentiable on [xl, xu]; using the representation of the solution g1 in

equation (25), we see that g1 ∈ C1,∞(I). To prove the iterative steps, let us assume that we know

that gk ∈ Cj,∞(I) for all j = 1, . . . , k; by assumption 2, the functions ξ1, ξ2, ξ3, ξ4 are infinitely

differentiable on [xl, xu], using the representation of the solution gk+1 in equation (31) we see that

gk+1 ∈ Ck+1,∞(I); this ends the iterative step.

We conclude that gk ∈ Ck,∞(I) ⊂ C1,2(I) for all k = 1, 2, . . .. Since the sum of the functional

series converges, this shows that gp ∈ C1,2(I). By construction all gk(T, ·) = 0 and so gp ∈ G.

Proof of Theorem 11. By assumption, gp = χ ln ε +∑∞

k=1 pkgk for |p − 1| < p. Based on

equation (16) this implies for |p− 1| < p that ϕp =∑∞

k=1 pkϕk using the terms defined in equation

(22). Similarly, based on equation (16) we have ϕp = δψεη +∑∞

k=1 pk ck using the terms defined in

equation (22). Taking limits n→ ∞ of Gn,Φn,Γn proves the statement.

Proof of Propositions 12 and 13. We start proving the statement of propositions 12 for

gk for all k = 1, 2, . . .; the proof is inductive. For this, we note first that the statement is true for

k = 1, see equations (30, 36). For the inductive step, assume that for all 1 ≤ i ≤ k functions hg,i

exist such that for (t, x) ∈ I : gi(t, x) = hg,i(x)(T − t)i. Then, for all j = 1, . . . , k − 1, we have

∂gk∂x

=∂hg,k∂x

(T − t)k,∂2gk∂x2

=∂2hg,k∂x2

(T − t)k,∂gj∂x

∂gk−j∂x

=∂hg,j∂x

∂hg,k−j∂x

(T − t)k. (B-8)

Also, we have in consumption-savings problems for n = 1, . . . , k:

∑j1,...,jn≥1j1+...+jn=k

∫ T

t

n∏m=1

gjmdt =∑

j1,...,jn≥1j1+...+jn=k

∫ T

t

n∏m=1

hg,jm(T − t)jmdt

=∑

j1,...,jn≥1j1+...+jn=k

n∏m=1

hg,jm

∫ T

t(T − t)kdt =

∑j1,...,jn≥1j1+...+jn=k

n∏m=1

hg,jm(T − t)k+1dt. (B-9)

Equations (31, 38) characterize gk+1 based on gi for all i = 1, 2, . . . , k and we conclude using the

above decompositions the stated separability for gk+1. This ends the inductive step and proves the

separability for all gk for all k = 1, 2, . . ..

Based on the defining equation (22) we then conclude that ϕk is separable into the product

of a function in the state variable x and a power function of order k in the remaining lifetime

T − t. Similarly, in consumption-savings problems, we conclude based on the defining equation (22)

together with equations (B-8, B-9), that ck is separable into the product of a function in the state

variable x and a power function of order k in the remaining lifetime T − t.

To prove proposition 13 for (gk)k, we proceed in the same way by induction and use the above

representations. First, we note that g1 fulfills the base function representation in this proposition,

see equations (25, 26). A careful review of equations (25, 26) along the above lines carries out the

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inductive step in the proof. Finally then, equation (22) concludes the statement for all functions

ϕk and ck.

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