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A number of empirical and theoretical studies have documented that jump risk has asubstantial impact on portfolio selection. Given that jumps are inherently infrequent, itis di¢ cult to estimate jump models with adequate precision. This paper presents a novelapproach to the optimal portfolio selection problem in a potentially large nancial marketfor an investor who faces both di¤usion and jump risk and who is averse not only to riskof loss but also to the uncertainty associated with jumps. More speci cally, we develop apathwise optimization procedure based on martingale methods and minimax results to solvefor the probability of the worst scenario and for the optimal portfolio strategy in a jump-di¤usion model. More importantly, our method avoids the curse of dimensionality and hencesigni cantly helps to solve a portfolio selection problem in a model with jump risk for aninvestor with ambiguity aversion. Finally we apply our theoretical results to another modelto examine the properties of the optimal portfolio choices. In striking contrast to a puredi¤usion model, our model indicates that the ambiguity aversion of an investor with regardto jump parameters may not reinforce the investors risk aversion.
Citation preview
Duality approaches to robust portfolio choice with ambiguity
aversion to jump risks
by
Xing Jin�
Warwick Business School, University of Warwick
Xudong Zeng
School of Finance, Shanghai University of Finance and Economics
�Corresponding author
1
Abstract
A number of empirical and theoretical studies have documented that jump risk has a
substantial impact on portfolio selection. Given that jumps are inherently infrequent, it
is di¢ cult to estimate jump models with adequate precision. This paper presents a novel
approach to the optimal portfolio selection problem in a potentially large �nancial market
for an investor who faces both di¤usion and jump risk and who is averse not only to risk
of loss but also to the uncertainty associated with jumps. More speci�cally, we develop a
pathwise optimization procedure based on martingale methods and minimax results to solve
for the probability of the worst scenario and for the optimal portfolio strategy in a jump-
di¤usion model. More importantly, our method avoids the curse of dimensionality and hence
signi�cantly helps to solve a portfolio selection problem in a model with jump risk for an
investor with ambiguity aversion. Finally we apply our theoretical results to another model
to examine the properties of the optimal portfolio choices. In striking contrast to a pure
di¤usion model, our model indicates that the ambiguity aversion of an investor with regard
to jump parameters may not reinforce the investor�s risk aversion.
JEL Classi�cation: G11
Key Words and Phrases: Ambiguity aversion, Optimal portfolio selection, Duality method
2
1 Introduction
Empirical evidence in �nance generally suggests that asset returns are not normally dis-
tributed and that the jump risk needs to be captured to model the skewness and the kur-
tosis of asset return processes; see Bakshi, Cao and Chen (1997), Bates (2000), Eraker,
Johannes and Polson (2003), among others. A number of empirical and theoretical stud-
ies have been demonstrated that jump risk has a substantial impact on portfolio selection;
see Liu, Longsta¤ and Pan (2003), and Das and Uppal (2004), for example. However, it
is di¢ cult to estimate jump models with adequate precision because jumps are inherently
infrequent. Our objective in this paper is to study the portfolio selection problem in multi-
asset and multi-state-variable models, where an investor faces both di¤usion risk and jump
risk and where the investor is averse not only to risk of loss but also to the model uncertainty
caused by jumps.
Prompted by the seminal work of Merton (1969, 1971) and Samuelson (1969), the dy-
namic portfolio selection problem has typically been studied in conjunction with continuous-
time models, primarily due to their analytical tractability. A standard assumption of
continuous-time models is that asset prices follow di¤usion processes. However, recent stud-
ies of the portfolio selection have demonstrated that optimal portfolios held by an investor
who faces jump risk are dramatically di¤erent from optimal portfolio in the absence of jump
risk. In other words, the economic loss associated with ignoring jump risk may be substan-
tial. For example, in a single-stock double-jump model, Liu, Longsta¤ and Pan (2003) �nd
that an investor is less willing to take leveraged or short positions than in a standard di¤u-
sion model due to the investor�s inability to hedge jump risks through continuous balancing.
In an international market setting, Das and Uppal (2004) investigate the e¤ect of jumps on
international portfolio selection. In their model, jumps tend to occur simultaneously across
3
countries because jumps act as contagions. They �nd that jumps reduce the gain from in-
ternational diversi�cation and that leveraged portfolios may incur large losses when jumps
occur. In all of these studies and in the rational expectation approach, model uncertainty is
excluded a priori because the agent is assumed to have precise information; accordingly, the
model parameters are taken to be estimated with in�nite precision. Because only discrete
data sets are available for continuous-time models, jumps are di¢ cult to identify and hence,
a jump model is di¢ cult to estimate with adequate accuracy. The model associated with the
point estimate, called the reference model, has a high likelihood of being wrong. Naturally,
an investor may be averse to model uncertainty (known as ambiguity aversion) and, as a
result, may make conservative portfolio choices to ensure that the chosen investment strat-
egy will perform well when a set of relevant models are used, including the reference model
and other competing models that are di¢ cult to distinguish statistically from the reference
model.
Ambiguity aversion or Knightian uncertainty is originally formulated by Gilboa and
Schmeidler (1989) under an axiomatic framework in static models with the preferences of the
investor characterized by a max-min expected utility. Recently, two approaches have been
developed that incorporate ambiguity aversion into dynamic settings. The �rst, known as
the multi-prior approach, is laid out in Epstein and Schneider (2003) and Chen and Epstein
(2002), where investor preference can be represented by a recursive max-min expected utility
over a set of multi-prior probabilities. This formulation leads to dynamically consistent port-
folio rules in dynamic asset allocation problems. The second approach, called robustness, is
pioneered by Hansen and Sargent (1995) based on the application of robust control theory to
economic problems. Major developments in this approach can be found in Hansen, Sargent
and Tallarini (1999) and Anderson, Hansen and Sargent (2003). Both approaches have been
4
used to model the aversion to model uncertainty in economics and �nance. In particular,
Maenhout (2004) applies the approach developed by Anderson, Hansen and Sargent (2003)
to a dynamic asset allocation problem in a pure-di¤usion model in which an investor allo-
cates his or her wealth between a risky stock and a risk-free bond. Studying the e¤ects of
model uncertainty on both dynamic portfolio rules and equilibrium asset pricing, Maenhout
�nds that robustness not only increases the investor�s risk aversion, but also increases the
equilibrium equity premium and lowers the risk-free rate. By extending the results obtained
in Anderson, Hansen and Sargent (2000) to a one-stock jump-di¤usion setting, Liu, Pan
and Wang (2005) study the asset pricing implications of imprecise knowledge about jumps
and demonstrate that uncertainty aversion to jumps can explain the volatility smirk pattern
observed in the literature on option pricing. In the present paper, we consider a multi-asset
jump-di¤usion model and incorporate model uncertainty with respect to jumps by using the
approach to ambiguity aversion employed in Liu, Pan and Wang (2005). As is well under-
stood, �nding the solution to an optimal portfolio selection problem in an incomplete market
in which there are a large number of assets and state variables, especially when model un-
certainty is present, is extremely di¢ cult. In fact, Bardhan and Chao (1996) demonstrate
that a market with unpredictable jumps is inherently incomplete regardless of the number
of basic traded assets. In contrast, an incomplete pure-di¤usion model can be completed
by increasing the number of traded stocks. In general, explicitly solving the corresponding
portfolio selection problem in an incomplete market and determining the probability of the
worst case scenario are daunting tasks. One usually uses either the HJB equation or the
duality-martingale method. As is well known in �nance, it is di¢ cult to apply the HJB
equation to a high-dimensional problem due to the curse of dimensionality, and it is also
challenging to use the martingale method in an incomplete market because there are in�-
5
nitely many martingale measures. In the present paper, we develop a new approach based
on the martingale methods and minimax results to evaluate the probability of the worst case
scenario and then solve the corresponding portfolio selection problem.
Our method has several attractive features. First, we solve a pathwise minimization
problem instead of a minimax problem over portfolio strategies and alternative probabilities
in the primal problem or a minimization problem over in�nitely many martingale measures in
the dual problem. As a result, this method dramatically simpli�es the process of solving the
portfolio selection problem under ambiguity aversion to jump risk. Second, we reduce a high-
dimensional portfolio selection problem into a set of two-dimensional optimization problems,
so that we can solve the original dynamic portfolio selection problem in jump-di¤usion models
with a large number of assets and state variables and with ambiguity aversion to jump risk
in a computationally e¢ cient way.
Our paper is related to the work of Jin and Zhang (2012) in that they use a decomposition
approach to solve a portfolio selection problem that may include a large number of assets and
state variables with ambiguity aversion to jump risk. However, Jin and Zhang do not provide
a simple method of determining the probability of the worst outcome. These researchers�
approach is based on the HJB equation for CRRA utility functions and is not easy to extend
to more general HARA utility functions. The methods used in the present paper di¤er from
this earlier approach in that we develop a pathwise optimization method based on a duality-
martingale approach in combination with minimax results. Thus, our approach can be used
with more general HARA utility functions.
Our paper is also similar to the work of Das and Uppal (2004) and Ait-Sahalia, Cacho-
Diaz and Hurd (2009). These researchers solve the portfolio selection problems for jump-
di¤usion models. However, in their models, there is only one type of jump, and there is no
6
state variable and no model uncertainty. In contrast, we obtain tractable solutions to indicate
optimal portfolio strategies under a jump-di¤usion model that includes a large number of
assets and state variables and that incorporates model uncertainty.
The rest of the paper is organized as follows. In the next section, we present the frame-
work for Merton�s dynamic portfolio selection problem and demonstrate how it can be ex-
tended using ambiguity aversion. In Section 3, we develop a pathwise optimization approach
using the martingale methods and minimax results. Section 4 uses the duality method to
solve the dynamic portfolio choice problem in a two-stock model and evaluates the optimal
portfolio strategies. Section 5 concludes the paper. All proofs are collected in the appendices.
2 Merton�s problem and ambiguity aversion
In this section we formulate a model of incomplete �nancial markets in a continuous time
economy where asset prices follow a multidimensional jump-di¤usion process on the �xed
time horizon [0; T ], 0 < T <1.
We consider a complete probability space (;F ; P ), where is the set of states of na-
ture with generic element !, F is the �-algebra of observable events and P is a probabil-
ity measure on (;F). The uncertainty of the economy is generated by a d-dimensional
standard Brownian motion BS(t) = (BS1 (t); :::; B
Sd (t))
0 , an l-dimensional standard Brown-
ian motion BX(t) = (BX1 (t); :::; B
Xl (t))
0 and an (n � d)-dimensional multivariate Poison
process denoted by N(t) = (N1(t); :::; Nn�d(t))0, which are all de�ned on the probability
space (;F ; P ), with Nk(t) denoting the number of type k jumps up to time t. Assume
BS(t) and BX(t) are correlated and have d� l correlation matrix �t. The �ow of information
in the economy is given by the natural �ltration, i.e., the right-continuous and augmented
�ltration fFtgt2[0;T ] = fFSt _ FX
t _ FNt ; t 2 [0; T ]g, where FS
t = �(BS(s); 0 � s � t),
7
FXt = �(BX(s); 0 � s � t) and FN
t = �(N(s); 0 � s � t). We suppose that observable
events are eventually known, i.e., F = FT .
We use a l-dimensional vector Xt = (X1t; :::; Xlt)0 to denote the state variables of the
economy. The state variables Xt may include stochastic volatilities and stochastic interest
rates as its components. For analytical tractability to be illustrated in Appendix B, we
assume that the state variables Xt follow a pure di¤usion process
dXt = bx(Xt)dt+ �x(Xt)dBX(t)
where bx(Xt) is an l-dimensional vector function and �x(Xt) is an l � l matrix function of
Xt, respectively. It should be noted the speci�cation of Xt excludes jumps in volatility.
As in Chapter VIII of Bremaud (1981), we assume that Nk admits stochastic intensity
�k(Xt), and the amplitude of the type k jump, denoted by Yk, has probability density
�k(t; dx), where �k(Xt) represents the rate of the jump process at time t, �k(t; dx) is Ft-
predictable and denotes the probability of getting a jump size x if there is a jump at time
t. For any two n-dimensional vectors x = (x1; :::; xn) and y = (y1; :::; yn), we denote the
component-wise multiplication as x � y = (x1y1; :::; xnyn).
We are now in a position to describe asset price processes. The market considered in this
paper includes m + 1 assets traded continuously on the time horizon [0; T ]. One of these
assets, called the bond, has a price S0(t) which evolves according to the di¤erential equation
dS0(t) = S0(t)r(Xt)dt (1)
S0(0) = 1
The remaining m assets, called stocks, are risky; their prices are modelled by the linear
8
stochastic di¤erential equation
dSi(t) = Si(t�)(bi(Xt)dt+ �bi(Xt)dBS(t) + �qi (Xt)(Y � dN(t))
where i = 1; :::;m and Y = (Y1; :::; Yn�d). Here �bi(Xt) is the d-dimensional di¤usion coef-
�cient vector and �qi (Xt) is the (n � d)-dimensional jump coe¢ cient vector. In particular,
the Brownian motions represent frequent small movement in stock prices, while the jump
processes represent infrequent large shocks to the market.
We now turn to the portfolio selection problem. In this paper, we focus on the extended
Merton�s problem of maximizing the expected utility from the terminal wealth while incor-
porating ambiguity aversion. Speci�cally, we consider an investor with utility function U(x)
and endowed with some initial wealth w0, which is invested in the above-mentioned m + 1
assets. Let �(t) = (�1(t); :::; �m(t)) denote a trading strategy, where �i(t) is the proportion of
total wealth invested in the i-th risky asset held at time t and Ft-predictable. Any portfolio
policy �(t) has an associated wealth process Wt that evolves as
Wt = W0 +
Z t
0
r(s)Wsds+
Z t
0
Ws�(s)(b(s)� r(s)1m)ds
+
Z t
0
Ws�(s)�b(Xs)dBS(s) +
Z t
0
Ws��(s�)�q(Xs)(Y � dN(s))
where �b(Xt) is an m� (n� d) matrix with �bi being its i-th row, �q(Xt) is the m� (n� d)
matrix, with �qi being its i-th row. Here we use 1m to denote the m-dimensional column
vector of ones. A portfolio rule �(t) is said to be admissible if the corresponding wealth
process satis�es Wt � 0 almost surely. We use A(w0) to denote the set of all admissible
trading strategies. And we denote byW(w0) the family of all wealth processes generated by
admissible trading strategies in A(w0).
9
The traditional Merton�s problem without ambiguity aversion is that the investor at-
tempts to maximize the following quantity
u(w0) = maxW2W(w0)
J(w0) = E [U(WT )]
where the utility function U(x) is non-decreasing and concave on R = (�1;1):
Our next step is to incorporate ambiguity aversion into Merton�s problem. Suppose that
an investor fears the possibility of model mis-speci�cation and makes the decision to guard
against the worst case scenario. Given that rare events are typically high impact events and
that the parameters of underlying jump processes are di¢ cult to estimate with adequate
accuracy, we focus on the investor�s ambiguity aversion to uncertainty with regard to jump
parameters. In other words, the problem of the investor stems from a class of prior models
generated by imprecise estimates of jump parameters, e.g., jump intensity and expected
jump size. The investor considers the point estimates and the corresponding model (called
the reference model) to be the most reliable, but the investor also explicitly recognizes that
the competing models are di¢ cult to distinguish statistically from the reference model. As
a result, the investor makes a precautionary portfolio choice to guard against the competing
alternatives and to ensure that his or her portfolio strategy performs reasonably well even
when if the worst case scenario occurs. In the meantime, the select of any model other
than the reference model is penalized because the selection is a deviation from the reference
model.
Before de�ning the utility function that includes the ambiguity aversion and the deviation
penalty, we introduce a set of probability measures, denoted by P, that speci�es alternative
10
models of concern. Toward this end, we de�ne the martingale di¤erential as
q(dt; dx) = (q1(dt; dx); :::; qn�d(dt; dx))
where
qk(dt; dx) = dNk(t)� �k(Xt)�k(t; dx)dt
where k = 1; :::; n�d. Let P be the probability measure associated with the reference model.
Each probability measure P (�) 2 P has a Radon-Nikodym derivative, dP (�)dP
= � =Qn�d
k=1 �(k)t ,
with respect to P , where �(k)t is modelled by the stochastic di¤erential equation
�(k)t = �
(k)0 +
Z t
0
ZAk
(#k(s) k(s; x)� 1)�(k)s�qk(ds; dx) (2)
where �(k)0 = 1; #k(s) and k(s; x) are positive stochastic processes satisfying the following
relationship ZAk
k(t; z)�k(t; dz) = 1
for k = 1; :::; n � d. Here Ak is the support of size of the k-th jump. In particular, we set
Ak = (0;1) for a positive jump, Ak = (�1; 0) for a negative jump, and Ak = (�1;1) for
a mixed jump. From now on, we suppress the dependence of �k(Xt); #k(t); �k(t; dz) and
k(t; z) on t and Xt for notational convenience.
By Theorem T10 of Bremaud (1981), under the probability measure P (�), the k-th jump
intensity �k and density function�k(dz) are changed into #k�k and k(z)�k(dz), respectively.
To better understand this modelling, we consider a case where the investor is only averse to
the intensity of the �rst jump and is comfortable with the estimates of other parameters, then
#j = 1 for j = 2; ::; n� d, and j(z) = 1 for j = 1; :::; n� d. Thus, �(k)t = 1; k = 2; :::; n� d;
11
and �(1)t de�ned by (2) is reduced to
�(1)t = �
(1)0 +
Z t
0
ZA1
(#1 � 1)�(1)s�q1(ds; dx):
In other words, all jump distributions and jump intensities except for the �rst jump in-
tensity in an alternative model are the same as those in the reference model whereas
the intensity of the �rst jump becomes #1�1 in the alternative model. Furthermore, let
�1 denote the con�dence interval of one standard deviation of estimate of �1, that is,
�1 = [b�1 � std(b�1); b�1 + std(b�1)], where b�1 is a point estimate of �1 and std(b�1) is thecorresponding standard deviation. Then, the set of all possible values of #1 is the interval
[(b�1 � std(b�1))=b�1; (b�1 + std(b�1))=b�1].In the remainder, we use �k to denote the set of all possible values of #k(t), which is
associated with the con�dence interval of a point estimate of �k. For the k-th jump size, we
use k to denote the set of all possible nonnegative functions of k(t; z) given by
k =
� k : k � 0;
ZAk
k(t; z)�k(t; dz) = 1
�: (3)
In general, we let � = �1 � �2 � � � � � �n�d and = 1 � 2 � � � � � n�d and we let P
denote the set of all alternative probabilities determined by � and :
We now de�ne the utility function. Following Liu et al. (2005), we make some changes to
Merton�s problem (described above). We begin by formulating a utility function in a discrete-
time setting and then, by taking the limit, derive the utility function for a continuous-time
model. For illustrative purposes, our penalty is the standard measure of entropy, which is
a special instance of the penalty function used in Liu et al. (2005). However, it is easy to
extend the results shown in the present paper to Liu et al.�s more general penalty function.
12
To be more speci�c, for a �xed time period �t, the time-t utility is recursively given by
Ut = infP (�)2P
(��E�t (Ut+�t)
� n�dXk=1
1
'kE�t
"ln
�(k)t+�t
�(k)t
!#+ E�
t (Ut+�t)
)(4)
with UT = U(WT ), and E�t denoting the conditional expectation under the probability P (�).
As in Liu et al. (2005), E�t
hln��t+�t�t
�imeasures the discrepancy between probabilities P (�)
and P , which is the standard measure of entropy. The coe¢ cient 'k represents the ambiguity
aversion to the kth jump. The minimization problem re�ects aversion to ambiguity of the
investor who worries about the imprecise estimation of parameters. Therefore, the investor
makes decisions to guard against the worst scenario. �(x) is a normalization factor and, for
tractability, we assume �(x) = (1� )x as in Maenhout (2004). In Proposition 1 given below,
by letting �t tend to zero, the continuous-time version Ut of utility (4) will be derived. Let
J(t;Wt; Xt) denote the indirect utility function given by
J(t;Wt; Xt) = supW2W(w0)
fUtg: (5)
Following Merton (1971), using the standard approach to stochastic control and an appro-
priate Ito�s lemma for jump-di¤usion processes, we can derive the optimal portfolio weights,
�, and the corresponding indirect value function, J , of the investor�s problem following the
HJB equation below:
0 = max�
�Jt +
1
2W 2�T�b�
Tb �JWW +W [�T(b(t)� r1m) + r]JW (6)
+bx(t)JX +W�T�b�Tt �
xT(t)JWX +1
2Tr(�x(t)�xT(t)JXXT)
+ infP (�)2P
(n�dXk=1
E� [J(W +W�T�qkz)� J(W )] +H(�t)J
))
13
where �qk denotes the k-th column of �q and the function H(�t) is given in Proposition 1.
We use T to denote the transpose transformation.
As is well understood, the preference de�ned in (4) is dynamically consistent because it
is de�ned recursively; see Epstein and Schneider (2003) and Wang (2003). In the rest of the
paper, we assume that the utility function U(x) is non-positive, which includes the CRRA
and HARA utility functions in Section 3 as special cases.
Proposition 1 The continuous-time version of utility satisfying equation (4) is given by
Ut = E�t
heR Tt HsdsU(WT )
i(7)
where
Ht = H(�t) = (1� )n�dXk=1
�k'k
ZAk
[#k(t) k(t; z) ln(#k(t) k(t; z)) + 1� #k(t) k(t; z)]�k(dz)
with H(�t) � 0.
Furthermore, suppose that for some � > 1;
sup k2k
ZAk
�k (s; z)�k(dz) <1;8s 2 [0; T ]: (8)
Then
J(t;Wt; Xt) = supWfUtg = inf
�supW
E�t
heR Tt HsdsU(WT )
i: (9)
Proof. See Appendix A.
The new form of utility function (9) has an attractive feature. The maximization problem
in the "inf sup" problem, which is given by the second equality, is an investment optimization
14
problem. Thus, the new expression makes it possible to use the duality method developed
by Kramkov and Schachermayer (1999, 2003) and Schied and Wu (2005) to evaluate the
optimal expected utility function given by (5). In general, it is much more di¢ cult to solve
the original "sup inf" problem de�ned by the �rst equality in (9).
Before concluding the section, we consider the speci�cation of jump distribution and its
Radon-Nikodym derivative in Liu, et al. (2005). According to their equation (1), the rate of
endowment �ow fYt; 0 � t � Tg solves the stochastic di¤erential equation
dYt = �Ytdt+ �YtdBt + (eZt � 1)Yt�dNt;
where Y0 > 0; � � 0 and � > 0 are constants. Here B denotes a standard Brownian motion
and N is a Poisson process with intensity � > 0. The jump amplitude is controlled by Zt,
which is normally distributed with mean �J and standard deviation �J .
Following the equation (2) in Liu, et al. (2005), the Radon-Nikodym derivative is given
by
d�t = (ea+bZt�b�J� 1
2b2�2J � 1)�t�dNt � (ea � 1)�tdt;
where a and b are predictable processes. Then, according to Liu, et al. (2005), the density
function of the jump size is changed into (t; z)�(dz) under the probability P (�), where
�(z) is the distribution function of jump size eZt � 1 under probability P and (t; z) =
(1 + z)be�b�J�12b2�2J . Then, we can verify that for � > 1;
ZA
�(z)�(dz) = e12�(��1)b2�2J :
Assume the set is the convex hull of functions (t; z) = (1 + z)be�b�J�12b2�2J . Then, the
15
condition (8) is satis�ed provided that predictable processes b are bounded.
3 Duality approach to solving the indirect utility function
In this section, we develop a pathwise optimization procedure that can be used to solve the
portfolio problem formulated in Section 2. Speci�cally, we use the duality method developed
in Kramkov and Schachermayer (1999, 2003), and Schied and Wu (2005), together with the
minimax theorem in Fan (1953) and Proposition 1 of the present paper. For tractability,
we �rst consider a CRRA utility function; then, we extend our results to more general
HARA utility functions. As noted in the introduction, Bardhan and Chao (1996) show that
once unpredictable jumps are included in the model, the market is inherently incomplete,
regardless of whether m � n or m < n. In contrast, in a pure-di¤usion economy, increasing
the number of traded assets can always complete the market. For tractability, we consider
the case of m � n, in which the number of risky assets is greater than or equal to the sum
of the di¤usions and jumps. Propositions 3 and 4 indicate that our methods are especially
powerful in these cases. For the case m < n, Jin and Zhang (2012) adopt the ��ctitious
completing" approach developed by Cvitanic̀ and Karatzas (1992) to show that solving the
portfolio selection problem in the original market can be converted into solving one in a
set of �ctitious markets. In particular, the number of risky assets is equal to the sum of
the di¤usions and jumps, that is, m = n in each �ctitious market, and hence, the results
developed in the present paper can be used to solve the optimal portfolio selection problem
in each �ctitious market.
For a market with asset returns that are consistent with the aforementioned jump-
di¤usion processes, Bardhan and Chao (1996) argue that if m > n and there are no arbitrage
opportunities, then m � n assets in the market are redundant and can be removed. This
16
scenario is similar to the case in a pure-di¤usion economy. In the remainder of the present
paper, we consider the case in which m = n, although our method is more generally ap-
plicable to the case in which m � n; there are more assets than di¤usions and jumps, or
otherwise, the number of assets equals the number of di¤usions and jumps. Without loss of
generality, we assume that the Brownian motions underlying the asset returns and the state
variables are identical, namely, BS(t) = BX(t) = B(t).
3.1 CRRA utility function
In this section, we consider the constant relative risk aversion (CRRA) utility function given
by
U(x) =
8>><>>:x1�
1� ; 8x > 0
�1; 8x � 0: (10)
For practical relevance, we assume the relative risk aversion coe¢ cient is greater than one.
In order to calculate J(t;Wt; Xt) de�ned by (5), we now lay out necessary notation. As in
Section 2, we use P (�) to denote the probability de�ned by the Radon-Nikodym derivative
� given by (2) with (#1(t); :::; #n�d(t)) and ( 1(t; z); :::; n�d(t; z)). We let Q� be the family
of all densities of equivalent local martingale measures with respect to the probability P (�).
We use E�(�) to denote the expectation under P (�). According to the discussion in the
previous section, the jump intensities and jump size distributions under P (�) are given by
��k = #k(t)�k and ��k(dz) = k(t; z)�k(dz), respectively, k = 1; :::; n� d:
We now introduce a characterization result of Q� developed in Bardhan and Chao
(1996). To this end, we assume the matrix � = [�b;�q] is invertible. Here we use e� =
17
(e�b1; :::;e�bd;e�q1; :::;e�qn�d)0 to denote the market price of risk given by
e� =0BB@ e�be�q1CCA = ��1(b(t)� r1n+�q�
� � �): (11)
Note that
��1�q =
0BB@ 0d�(n�d)
I(n�d)�(n�d)
1CCA ;
where 0d�(n�d) denotes the d� (n�d) matrix of zeros and I(n�d)�(n�d) is the (n�d)� (n�d)
identity matrix. Hence, e� can be rewritten as
e� =0BB@ e�be�q1CCA =
0BB@ �b
�q + �� � �
1CCA ; (12)
where 0BB@ �b
�q
1CCA = ��1(b(t)� r1n):
Let �loc denote the family of triples � = (v; �; �), such that
v(t) = (v1(t); :::; vd(t))T
�(t) = (�1(t); :::; �n�d(t))T
�(t) = (�1(t; z); :::; �n�d(t; z))T
are predictable processes, � and � are strictly positive, � satis�es
ZE
�k(t; z)��k(dz) = 1
18
for t 2 [0; T ] and k = 1; :::; n� d, and the following equation holds:
�bv(t) + �q�� � (�� �(t) � e�) = b(t)� r1n+�q�
� � � (13)
or equivalently,
v(t) = e�b;�� � (�� �(t) � e�) = e�q (14)
where
� = (�1; :::; �n�d); e� = (e�1; :::; e�n�d) (15)
�k =
ZE
z��k(dz); e�k = ZE
z�k(t; z)��k(dz)
for t � 0 and k = 1; :::; n� d. For each � 2 �loc, de�ne the local martingale,
��(t) = �b�(t)�q�(t); (16)
where
�b�(t) = exp
��Z t
0
vT (s)dB(s)� 12
Z t
0
jjv(s)jj2ds�;
�q�(t) =
n�dYk=1
Nk(t)Yi=1
(�k(tki )�k(t
ki ; z
ki )) exp
�Z t
0
ZAk
(1� �k(s)�k(s; z))��k�
�k(dz)ds
�:
In particular, ��(t) is a supermartingale for each � 2 �loc since it is non-negative. We use �
to denote the subset of �loc for which ��(t) is a martingale.
The following lemma is one of the main results in Bardhan and Chao (1996) and plays a
19
key role in our paper.
Lemma 2 A measure Q 2 Q� if and only if there exists a triple � 2 �, such that the
Radon-Nikodym derivative dQdP= ��(t).
Proof. See Bardhan and Chao (1996).
Equipped with Proposition 1 and Lemma 2, we are able to solve the portfolio problem
by using the duality method developed in Kramkov and Schachermayer (1999, 2003) and
Schied and Wu (2005).
Proposition 3 Under the assumptions of Proposition 1, we have the following duality result
J(t;Wt; Xt) =W 1� t
1�
sup�sup�2Q�
E�t
he1
R Tt (Hs+(1� )r)ds��(t; T )
1� 1
i!
: (17)
And moreover,
sup�sup�2Q�
E�t
he1
R Tt (Hs+(1� )r)ds��(t; T )
1� 1
i= Et
"��b�(t; T )
�1� 1 exp
Z T
t
n�dXk=1
infck2C�k
TD�k;eck2 eC�k T ~D�
k
ZAk
gk(z; ck;eck)�k(dz)ds!#� f(t;Xt) (18)
where ��(t; T ) = ��(T )=��(t) and �b�(t; T ) = �b�(T )=�
b�(t). Here gk(z; ck;eck), C�k ; D�
k;eC�k and
eD�k are given in Appendix B.
Furthermore, the probability of the worst case is given by (48) in Appendix B.
Proof. See Appendix B.
In (18), we have translated the original optimization problem over the stochastic processes
� and � into a pathwise minimization problem. The former, as is well understood, is notori-
ously di¢ cult to solve due to in�nitely many Radon-Nikodym derivatives � and martingale
20
measures � and lack of a closed-form solution for the expectation E�t [�]: The latter is n � d
minimization problems over a subset in the two-dimensional real space R2 and are straight-
forward to solve. In the meantime, it is free of the curse of dimensionality caused by n� d,
the number of jumps, and thus, it can lead to signi�cant reduction in computation when
n� d is large. In short, the f(t;Xt) can be evaluated by the standard Monte Carlo method
in combination with the pathwise minimization problems. As a result, the optimal portfo-
lio strategy can be derived through a HJB equation satis�ed by the indirect value function
J(t;Wt; Xt) in (17), which will be obtained as a special case of the proposition in the next
section for general HARA utility functions.
In particular, for the case without the ambiguity, that is, � = 1, the equation (17) is
reduced to
J(t;Wt; Xt) =W 1� t
1�
�sup�2Q1
Et
�e1�
R Tt rds��(t; T )
1� 1
��
which is the indirect utility function of maximizing utility of terminal wealth in the incom-
plete market.
Then, according to equation (18), the above indirect utility function can be further
calculated as follows:
sup�2Q1
Et
he1�
R Tt rds��(t; T )
1� 1
i= Et
"��b�(t; T )
�1� 1 exp
n�dXk=1
Z T
t
ZAk
infckgk(z; ck)�k(dz)ds
!#
where gk(z; ck) can be derived in Appendix B as
gk(z; ck) = �ck�qk +1
"�1� 1
� �1�ckz + 1�
1
�1� � 1#�k:
21
3.2 HARA utility function
In this section, we use results developed in the last section to solve the optimal portfolio
choice problem in a model in which an investor has a HARA utility function but the bond
and stock prices remain unchanged. To be more speci�c, a HARA utility function is given
by
U(x) =
8>><>>:11� (x� b)1� ; 8x > b
�1; 8x � b
:
When b = 0; U(x) reduces to a CRRA utility function. Here we consider a realistic case
with 0 < b < W0; that is, the relative risk aversion is decreasing with x.
Proposition 4 Under the assumptions of Proposition 1, we have the following duality result
J(t;Wt; Xt) =(Wt � b�t)
1�
1�
�sup�sup�2Q
E�t
�exp
�1
Z T
t
(Hs + (1� )r)ds
���(t; T )
1� 1
�� =
(Wt � b�t)1�
1� (f(t;Xt))
;
where �t = Et
hexp
��R Ttrds��b�(t; T )
i; and f(t;Xt) is given by (18).
Proof. See Appendix C.
Before concluding this section, we provide the optimal portfolio rule given below. For
simplicity, we consider the reference model only since it is straightforward to extend the
result to the worse case model once the optimal � is obtained in Proposition 4.
Proposition 5 The optimal portfolio weight �� = (��1; :::; ��n) is given by
�� =�e��b1; :::; e��bd; e��q1; :::; e��q(n�d)���1
22
where
(e��b1; :::; e��bd)> = Wt � b�tWt
"e�b + �t�
x>fXf
#
and e��qk solves the following optimization problem:supe�qk2Fk e�qk(W � b�t)
� W (e�qk � �kak) +�k1�
ZAk
[W (1 + e�qkz)� b�t]1� �k(dz) (19)
for k = 1; :::; n� d, where the set Fk is de�ned in Appendix D.
Proof. See Appendix D.
4 The comparative statics of risk aversion and ambiguity aversion
In this section, we apply the theoretical results developed above to a two-stock jump-di¤usion
model in order to address two issues. First, we examine the investment behavior of an investor
in a multi-asset model when she faces more frequent jumps, with comparisons to those in
a single-stock market, as in Liu, et al. (2003). Second, by quantifying the sensitivity of
the optimal jump exposure to the risk aversion and the ambiguity aversion coe¢ cients, we
investigate whether ambiguity aversion to jumps a¤ects the investment behavior of investors
in the same way as increased risk aversion does.
For this, we specify a two-stock model. More precisely, we assume that stock prices follow
the jump-di¤usion processes below
dSi;t = Si;t�[(r + �i + �i�)dt+ �bidBt + �qiY dNt]; i = 1; 2;
where Bt is the standard Brownian motion and Nt is a one-dimensional Poisson process with
the jump intensity �. Here, for the ith stock we use �i and �i� to denote risk premium
23
components associated with the di¤usion and jump risks, respectively. For simplicity, we
assume that the jump size Y = exp(U) � 1 with U is normally distributed with mean �Y
and variance �2Y : In particular, if the jump size Y is deterministic, the stock price processes
reduce to the model in Section 4.1 of Liu, et al. (2003). Let � and �� denote the jump
intensities in the reference model and in the worst case, respectively. Similarly, we use b��qand e��q to denote the optimal exposure to the jump in the reference model and in the worstcase. In this model, the market price of risk can be expressed as
�t =
0BB@ �bt
�qt
1CCA =
0BB@ �b1;t + �b2;t�
�q1;t + �q2;t�+ �a
1CCA ;
where a is the expected jump size. We assume the investor has a CRRA utility function
de�ned by (10). Our static analysis is based on the following result.
Proposition 6 Let e��q and �� be interior optimal solutions to the maxmin problem (56) in
Appendix E. Then, we have
�� = � exp
�'
�1
1�
Z 1
�1[1� (1 + e��qz)1� ]�(dz)� �q2e��q�� : (20)
@e��q@
= �
Z 1
�1z(1 + e��qz)� ln(1 + e��qz)�(dz)
Z 1
�1z2(1 + e��qz)� �1�(dz) < 0; (21)
@e��q@��
= � �q1(��)2 A
; (22)
and
@e��q@'
= � �q1��B
(��)2A+ '(�q1)2; (23)
24
where
A =
Z 1
�1z2(1 + e��qz)� �1�(dz) > 0;
and
B =1
1�
Z 1
�1[1� (1 + e��qz)1� ]�(dz)� �q2e��q:
In particular,@e��q@'
> 0 and @��
@'< 0; if �q1 > 0 and
�q2 >1
1�
�1� exp
�(1� )�Y +
(1� )2�2Y2
��: (24)
Proof. See Appendix E.
To better understand an investor�s investment behavior in a multi-asset jump-di¤usion
model with and without ambiguity aversion, we �rst focus on the case without ambiguity
aversion and examine the impact of the jump intensity � on the optimal jump exposure
b��q and portfolio strategy �� = (��1; ��2). Speci�cally, we turn o¤ the ambiguity aversion
by letting � = 0: Then �� = � and e��q = b��q. From Proposition 5, b��q solves the followingone-period portfolio choice problem with CRRA utility function:
supb�q2[0;1) b�q(�q � �a) +�
1�
ZA
[(1 + b�qz)1� � 1]�k(dz): (25)
The optimal portfolio rule in Section 4.1 of Liu, et al. (2003) solves the problem (25),
implying that the optimal portfolio rule in the single-stock model is the same as the optimal
jump exposure b��q in the two-stock model but portfolio strategies in terms of response tojumps may di¤er. Speci�cally, consider the case where �q1 > 0, which corresponds to the case:
� > 0 in Section 4.1 of Liu, et al. (2003). From (22),@b��q@�
< 0, namely, the investor will reduce
her optimal jump exposure when facing more frequent jumps. By contrast, the investor may
25
increase her holding in one stock while reduce her holding in another stock. This can be seen
from �� = (��1; ��2) = (�
�b ; �
�q)�
�1 with ��b = (�b1;t+ �
b2;t�)= . This is in contrast to the single-
stock jump-di¤usion model considered in Liu, et al (2003), where they demonstrate that a
risk averse investor is fearful of both negative and positive jumps by reducing the magnitude
of her position (jump exposure in the single-stock model) in the stock with increasing jump
intensity, despite more positively skewed returns caused by positive jumps. The reason is
that, in a single-stock model, there is no bene�t of diversi�cation and the portfolio�s exposure
to di¤usion is proportional to that to jumps, and hence, a reduction in exposure to jump will
lead to reduction in exposure to di¤usion. As a result, the investor will reduce investments
with sharp unforeseeable movements. In contrast, an investor can choose the di¤usion and
jump exposures e�b and e�q separately in a multi-stock model to diversify risk, and, in general,the portfolio�s exposure to di¤usion is no longer proportional to that to jumps. Furthermore,
the exposure to a jump is the optimal portfolio weights in a one-period pure jump model and
is determined by the problem (25). And hence, the investor behaves myopically. As a result,
the optimal portfolio weight in the pure-jump market (or exposure to the jump in the original
market) is positively related to the mean-variance portfolio, which is mv(�) = c��q1�+ �q2
�for some c > 0 in this case. Obviously, mv(�) decreases with � if �q1 > 0.
We now turn to the comparison of impacts of risk aversion and ambiguity aversion to the
jump intensity on the jump exposure. Equation (21) says that the jump exposure decreases
with the risk aversion coe¢ cient : By contrast,@e��q@�
> 0 suggests that the jump exposure
increases with the ambiguity aversion coe¢ cient �; implying the ambiguity aversion does not
reinforce the risk aversion. This is in striking contrast to the �ndings of Maenhout (2004),
Uppal and Wang (2003) and Liu (2011) in pure di¤usion models. To put it into perspective,
26
let us consider the investor�s one-period utility function without penalty in Appendix E
D(�q; e�) = e�q(�q1 + �q2e�) + 1
1� e�Z
E
[(e�qz + 1)1� � 1]�(dz):From the proof in Appendix E, the (24) implies that given e�q; the utility function D(�q; e�) isincreasing with the jump intensity e�: And thus, more ambiguity averse investor worries aboutsmaller jump intensity e�, namely, less frequent jumps, or equivalently, @��
@�< 0. In other
words, once jump risk is su¢ ciently compensated, that is, �q2 is big enough as indicated
by (24), the ambiguity averse investor worries about less frequent jumps instead of more
frequent jumps. This suggests that a more ambiguity averse investor in the one-period
market adjust the excess expected return �q1;t + �q2;t� downward by reducing � in the worst
case model. This is similar to the ambiguity aversion in a pure-di¤usion model investigated
in Maenhout (2004), in which the expected return of stock in the worst case is less than
the one in the reference model. And moreover, according to (22), given a smaller �, the
investor increases the jump exposure e��q since @e��q@�� < 0 given �q1 > 0: As a result, a more
ambiguity averse investor has a bigger jump exposure. Intuitively, in pure di¤usion models,
ambiguity aversion leads to an adjustment of the drift term in stock returns but leaves higher
moments unchanged, so it is observationally equivalent to increasing risk aversion. In our
paper, we study the aversion to model uncertainty associated with jump parameters. With
a jump-di¤usion process, the ambiguity aversion to jump related uncertainties leads to an
adjustment in higher moments. More precisely, the imprecise estimation of jump intensity
a¤ects the �rst and higher moments of stock returns. As a result, the ambiguity aversion
coe¢ cient � a¤ects the �rst and higher moments of stock returns. And furthermore, an
investor with a CRRA utility takes all moments into account when making optimal portfolio
27
choice and hence risk aversion and ambiguity aversion a¤ects the investment behavior in
di¤erent ways.
5 Conclusion
Solving the optimal dynamic portfolio selection problem for an incomplete market with or
without model uncertainty is a daunting task due to the curse of dimensionality. This paper
proposes a novel approach to the intertemporal portfolio selection problem in jump-di¤usion
models where the investor is averse not only to risk but also to model uncertainty. More
speci�cally, based on the duality-martingale method and the minimax theorem, we evaluate
the probability of the worst case scenario and the indirect value function by solving a pathwise
optimization problem. Then, the optimal portfolio rule can be readily obtained via the HJB
equation. One appealing feature of our approach is that our method can be used to consider
a large number of assets and state variables in a model with ambiguity aversion to jump
risk. Our approach also circumvents the problem of dimensionality.
Finally we use a two-stock model to illustrate that unlike in the pure-di¤usion models
explored in the existing literature, ambiguity aversion to jump parameters may not a¤ect an
investor�s behavior in the same way as increased risk aversion.
A Proof of proposition 1
As in Jin and Zhang (2012), we have
E�t
"ln
�(k)t+�t
�(k)t
!#=
1
�(k)t
Et[�(k)t+�t ln(�
(k)t+�t)� �
(k)t ln(�
(k)t )]:
28
Applying Ito�s lemma to the function f(x) = x lnx and the equation (2) gives
d[�(k)t ln(�
(k)t )] = �k
ZAk
[f(�(k)t� + �
(k)t� (#k(t) k(t; z)� 1))� f(�
(k)t� )
�f 0(�(k)t� )�(k)t� (#k(t) k(t; z)� 1))]�k(dz)dt
+
ZAk
[f(�(k)t� + �
(k)t� (#k(t) k(t; z)� 1))� f(�
(k)t� )]qk(dt; dz)
= �k
ZAk
[#k(t) k(t; z) ln(#k k(t; z)) + 1� #k(t) k(t; z)]�(k)t��k(dz)dt
+
ZAk
[f(�(k)t� + �
(k)t� (#k k(t; z)� 1))� f(�
(k)t� )]qk(dt; dz):
Hence
E�t
"ln
�(k)t+�t
�(k)t
!#=
1
�(k)t
Et[�(k)t+�t ln(�
(k)t+�t)� �
(k)t ln(�
(k)t )]
= �k
ZAk
[#k(t) k(t; z) ln(#k k(t; z)) + 1� #k(t) k(t; z)]�k(dz)�t:
We now turn to the proof of (7). Recall �(x) = (1 � )x and > 1: Then from the
results above, we have
��E�t (Ut+�t)
�E�t (Ut+�t)
n�dXk=1
1
'kE�t
"ln
�(k)t+�t
�(k)t
!#
= (1� )
n�dXk=1
�k'k
ZAk
[#k(t) k(t; z) ln(#k k(t; z)) + 1� #k(t) k(t; z)]�k(dz)�t
� H(�t)�t � Ht�t:
It is evident that H(�t) � 0 because x lnx+ 1� x � 0 for x > 0 and 1� < 0.
29
Hence,
��E�t (Ut+�t)
� n�dXk=1
1
'kE�t
"ln
�(k)t+�t
�(k)t
!#+ E�
t (Ut+�t)
= (1 +Ht�t)E�t (Ut+�t):
We consider the buy-and-hold strategy of investing all wealth in the bond at time 0. It
is su¢ cient to consider only those admissible dynamic strategies which are not worse than
the buy-and-hold strategy. We use W0(w0) to denote the set of wealth processes generated
by the such admissible dynamic strategies. In the sequel, we only consider wealth process
W 2 W0(w0) .
As a result, we shall have E�t [U(WT )] � E�
t [U(er(T�t)Wt)] � U(er(T�t)Wt) � U(Wt). In
particular E�T��t[U(WT )] � U(WT��t): Then, by noticing HT��t < 0 and U(WT��t) < 0;
for �t su¢ ciently small, we have
UT��t = inf�(1 +HT��t�t)E
�T��t(UT )
= inf�(1 +HT��t�t)E
�T��t [U(WT )]
� inf�(1 +HT��t�t)U(WT��t) � U(WT��t):
Similarly and backwardly, we can prove that 0 � Et[Us] � U(Wt) for s � t.
30
For a �xed W 2 W0(w0); let �� denote an optimal solution to the problem (4), then
Ut = (1 +Ht�t)E��
t (Ut+�t)
= E��
t (eHt�tUt+�t) + o(�t)E��
t (Ut+�t)
= E��
t (eHt�tUt+�t) + o(�t)U(Wt)
= E��
t (eHt�tE��
t+�t[eHt+�t�tUt+2�t]) + 2o(�t)U(Wt)
= E��
t (eHt�teHt+�t�tUt+2�t) + 2o(�t)U(Wt)
:::
= E��
t (e(Ht+Ht+�t+:::+HT��t)�tUT ) +
(T � t)
�to(�t)U(Wt)
! E��
t
heR Tt HsdsUT
i; as �t! 0: (26)
Therefore, we see that the continuous-time version of the utility function is given by
Ut = inf�E�t
heR Tt HsdsUT
i:
Next we show that
J(t;Wt; Xt) = supW
Ut = supWinf�E�t
heR Tt HsdsU(WT )
i= inf
�supW
E�t
heR Tt HsdsU(WT )
i:
Without loss of generality, we suppose n� d � 1; �k � �, Ak � A, that is, there is only one
type of jumps in the remainder of this proof.
31
Note
(1 +Ht�t)E�t (Ut+�t) = (1 +Ht�t)Et
��t+�t�t
Ut+�t
�
= (1 +Ht�t)Et
24e�(1�RA # (z)�(dz))�t N(�t)Yi=1
# (zi)Ut+�t
35= (1 +Ht�t)Et
he�(1�
RA # (z)�(dz))�tUt+�tjN(�t) = 0
ie���t
+(1 +Ht�t)Et
he�(1�
RA # (z)�(dz))�t# (z1)Ut+�tjN(�t) = 1
i(��t) e���t + o(�t)
= (1 +Ht�t)Et (Ut+�tjN(�t) = 0)
��Et��Z
A
# (z)�(dz)
�Ut+�tjN(�t) = 0
��t
+�e���tEt [# (z1)Ut+�tjN(�t) = 1]�t+ o(�t)
=
�1 +Ht�t� ��t
ZA
# (z)�(dz)
�Et (Ut+�tjN(�t) = 0)
+�e���tEt [# (z1)Ut+�tjN(�t) = 1]�t+ o(�t)
:= B(# ;W ; �t) + o(�t)
Note that the L1(�) =�x :RAjxjd�(z) <1
is a Hausdor¤ space under the weak topol-
ogy and the set de�ned by (3) is a compact set under the condition (8). Note that given
condition (8), 1+Ht�t���tRA# (z)d�(z) is positive for �t su¢ ciently small. Thus, it is
not di¢ cult to verify that B(# ;W ; �t) is a convex function of # and a concave function
of W . Then we can use Theorem 2 of the minimax result in Fan (1953) and the approach
used in the proof of Lemma 3.4 of Schied and Wu (2005) to get the following equality:
supWinf�B(# ;W ; �t) = inf
�supW
B(# ;W ; �t):
32
Hence
supWinf�
(��E�t (Ut+�t)
� n�dXk=1
1
'kE�t
"ln
�(k)t+�t
�(k)t
!#+ E�
t (Ut+�t)
)� sup
Winf�B(# ;W ; �t) = inf
�supW
B(# ;W ; �t)
� inf�supW
(��E�t (Ut+�t)
� n�dXk=1
1
'kE�t
"ln
�(k)t+�t
�(k)t
!#+ E�
t (Ut+�t)
):= inf
�supW
Vt:
Furthermore, in virtue of (26), we have
����supWinf�Vt � sup
Winf�E�t
heR Tt HsdsU(WT )
i����� sup
W
����inf� Vt � inf�E�t
heR Tt HsdsU(WT )
i����� sup
Wsup�
���Vt � E�t
heR Tt HsdsU(WT )
i���! 0 as �t! 0:
Hence
supWinf�Vt ! sup
Winf�E�t
heR Tt HsdsU(WT )
i; as �t! 0:
Similarly,
inf�supW
Vt ! inf�supW
E�t
heR Tt HsdsU(WT )
i:
Therefore, we have
J(t;Wt; Xt) = supW
Ut = supWinf�E�t
heR Tt HsdsU(WT )
i= lim
�t!0supWinf�Vt = lim
�t!0inf�supW
Vt = inf�supW
E�t
heR Tt HsdsU(WT )
i:
This complete the proof of (7). �
33
B Proof of proposition 3
We �rst prove the result (17). To this end, de�ne the convex conjugate of U(x):
V (y) = supx>0(U(x)� xy) =
1� y1�
1 :
From Proposition 1, the utility function can be rewritten as
Ut = E�t
�exp
�Z T
t
Hsds
�W 1� T
1�
�= DtE
�t
��(t; T )
W 1� T
1�
�
where Dt = E�t
hexp
�R TtHsds
�iand �(t; T ) =
exp(R Tt Hsds)Dt
. According to Schied and Wu
(2005),
J(t;Wt; Xt) = inf�Dt inf
y>0(v(y) +Wty)
where
v(y) = inf�2Q�
E�t
24�(t; T )V0@y ��(t; T ) exp
��R Ttrds�
�(t; T )
1A35 :Thus, by noticing > 1;
v(y) =
1� y1�
1 sup�2Q�
E�t
�exp
���1� 1
�Z T
t
rds
��(t; T )
1 ��(t; T )
1� 1
�
and consequently,
J(t;Wt; Xt) =W 1� t
1�
sup�sup�2Q�
E�t
�exp
�1
Z T
t
(Hs + (1� )r)ds
���(t; T )
1� 1
�!
;
completing the proof of (17).
The proof of (18) of Proposition 3 will be broken into several lemmas that are organized
34
into two subsections. We �rst evaluate sup�2Q� in (18). The Fenchel Duality Theorem plays
an important role in the proofs below. For more details about this theorem and relevant
notation, see Chapter 7 of Lunberger (1969). More speci�cally, the theorem states that
Let f be a proper convex function on set C and let g be a proper concave function on set
D. Suppose the regularity conditions are satis�ed, then the following equality holds
minx2C\D
(f(x)� g(x)) = maxx�2C�\D�
(g�(x�)� f ?(x�));
where f � is the convex conjugate of f on C�={x�jf �(x�) < 1g (also referred to as the
Fenchel-Legendre transform) and g� is the concave conjugate of g on D�={x�jg�(x�) > �1g.
That is,
f ? (x�) := sup fhx�; xi � f (x)jx 2 Cg ;
g� (x�) := inf fhx�; xi � g (x)jx 2 Dg ;
where hx�; xi will be de�ned later.
B.1 Auxiliary results for proof of (18)
Now we apply the Fenchel Duality Theorem to solve the following optimization problem:
supx2Xx�0
ZAk
�x1�
1 (z)�
�1� 1
�x(z)
���k(dz) (27)
subject to the constraint
ZAk
x(z)z��k(dz) =
ZAk
x(z)Sgn(z)��
k(dz) = ��qk��k
(28)
35
where
Sgn(z) =
8>>>>>><>>>>>>:�1; 8z < 0;
0; 8z = 0;
1; 8z > 0:
The constraint (28) is obtained from the equations (14) and (12), and
��
k(z) =
Z z
�1jsj��k(ds)
for k 2 f1; :::; n� dg. X is a linear normal space de�ned as follows.
X =
�x(z) :
ZAk
jx(z)j��k(dz) <1�
with norm
jjxjj =ZAk
jx(z)j��k(dz):
Then, the dual space X� of X is
X� = fx�(z) : x�(z) 2 L1(��k)g:
De�ne a concave function:
g0(x) =
8>><>>:x1�
1 �
�1� 1
�x; 8x � 0;
�1; 8x < 0:(29)
Then (27) is equivalent to the following problem:
supx2X
ZAk
g0(x(z))��k(dz)
36
subject to ZAk
x(z)Sgn(z)��
k(dz) = ��qk��k:
To employ the Fenchel Duality Theorem to solve the above problem, we lay out relevant
notation below. Set
D = X;
C =
(x 2 X :
ZAk
x(z)Sgn(z)��
k(dz) = ��qk��k
);
f(x) = 0;
g(x) =
ZAk
g0(x(z))��k(dz):
We �rst calculate the functional f � conjugate to f , given by
f �(x�) = supx2C[hx; x�i � f(x)] = sup
x2C
ZAk
x(z)x�(z)��
k(dz)
where
hx; x�i =ZAk
x(z)x�(z)��
k(dz); x 2 X and x� 2 X�:
Lemma 7 The conjugate space C� of f �(x�) is given by
C� = fx� : f �(x�) <1g = fcSgn(z) : c 2 Rg
and
f �(x�) = supx2C[hx; x�i] = �c�
qk
��k; for x� = cSgn(z) 2 C�:
37
Proof. De�ne a linear functional on X as
f0(x) =
ZAk
x(z)Sgn(z)��
k(dz)
and its zero space is given by
Ker(f0) =
�x 2 X : f0(x) =
ZAk
x(z)Sgn(z)��
k(dz) = 0
�:
Note that for any x(1) 2 Ker(f0), x(2) 2 C and integer N , Nx(1) + x(2) 2 C. Thus, we must
have hx(1); x�i = 0 in order that
f �(x�) = supx2C[hx; x�i] <1:
By Lemma 1 on Page 188 in Luenberger (1969), there exist a constants c, such that hx; x�i =
cf0(x) for any x 2 C. That is,
ZAk
x(z)x�(z)��
k(dz) =
ZAk
cx(z)Sgn(z)��
k(dz)
implying x�(z) = cSgn(z) and C� = fcSgn(z) : c 2 Rg. Moreover,
f �(x�) = supx2C[hx; x�i] = sup
x2C
ZAk
cx(z)Sgn(z)��
k(dz) = �c�qk��k;
completing the proof.
We now turn to the calculation of the functional g� conjugate to g. According to the
38
de�nition, for x� 2 X�,
g�(x�) = infx2D[hx; x�i � g(x)]
= infx2D
ZAk
[x(z)x�(z)jzj � g0(x(z))]��k(dz):
The conjugate space of g�(x�) is D� = fx� : g�(x�) > �1g. When using the Fenchel Duality
Theorem, we only need to calculate g�(x�) for x� 2 C�. To this end, we have the following
result.
Lemma 8 For x� = cSgn(z) 2 C�,
g�(x�) = infx2X
ZAk
[x(z)x�(z)jzj � g0(x(z))]��k(dz)
=
ZAk
infx2R[xx�(z)jzj � g0(x)]�
�k(dz)
= �1
�1� 1
� �1 ZAk
�cz + 1� 1
�1� ��k(dz): (30)
Proof. The inequality � is trivial, namely,
infx2X
ZAk
[x(z)x�(z)jzj � g0(x(z))]��k(dz)
�ZAk
infx2R[xx�(z)jzj � g0(x)]�
�k(dz):
We now prove �. We now solve the optimization problem
infx2R[xcjzjSgn(z)� g0(x)]
39
where g0 is de�ned in (29). It is easy to obtain the optimal solution as
bx = �cjzjSgn(z) + 1� 1
�� �1� 1
� (31)
and the corresponding optimal objective function is
infy2R[xcjzjSgn(z)� g0(x)]
= �1
�1� 1
� �1�cz + 1� 1
�1� :
If c < 1� 1 , bx 2 X and therefore, (30) holds true; if c = 1� 1
, then bx = (z + 1)� and we
will prove (30) still holds true. De�ne
xm(z) =
�z + 1 +
1
m
��
then xm 2 X and converges increasingly to bx as m tends to in�nity. Note that
g(xm) =
ZAk
[xm(z)x�(z)jzj � g0(xm(z))]�
�k(dz)
= ��1� 1
�1
m
ZAk
xm(z)��k(dz)�
1
ZAk
�z + 1 +
1
m
�1� ��k(dz)
� �1
ZAk
�z + 1 +
1
m
�1� ��k(dz):
Hence,
infx2X
ZAk
[x(z)x�(z)jzj � g0(x(z))]��k(dz)
� g(xm) � �1
ZAk
�z + 1 +
1
m
�1� ��k(dz):
40
and thus, by the Monotone Convergence Theorem,
infx2X
ZAk
[x(z)x�(z)jzj � g0(xm(z))]��k(dz)
� limm!1
"�1
ZAk
�z + 1 +
1
m
�1� ��k(dz)
#= �1
ZAk
(z + 1)1� ��k(dz)
= g(bx) = ZAk
infx2R[xcjzjSgn(z)� g0(x)]�
�k(dz);
completing the proof.
From the above lemma , we have
C� \D� =
(cSgn(z) :
ZAk
�cSgn(z)jzj+ 1� 1
�1� ��k(dz) <1
):
Without causing any confusion, we set
C� \D� =
(c :
ZAk
�cz + 1� 1
�1� ��k(dz) <1
): (32)
Consequently, by the Fenchel Duality Theorem, we can establish the following result.
Lemma 9
supx2C
ZAk
�x1�
1 (z)�
�1� 1
�x(z)
���k(dz)
= infc2C�\D�
"�c�
qk
��k+1
�1� 1
� �1 ZAk
�cz + 1� 1
�1� ��k(dz)
#:
41
Proof. By using the de�nition of functions f(x) and g(x),we obtain
supx2C
ZAk
�x1�
1 (z)�
�1� 1
�x(z)
���k(dz)
= supx2C[g(x)� f(x)]
= infc2C�\D�
[f �(x�)� g�(x�)]
= infc2C�\D�
"�c�
qk
��k+1
�1� 1
� �1 ZAk
�cz + 1� 1
�1� ��k(dz)
#;
with the second equality following the Fenchel Duality Theorem and completing the proof.
Before concluding this section, we present the following result of two properties of optimal
solution to the minimization problem above, which will be used in Appendix B.2.
Lemma 10 If c�k is the optimal solution to the problem in Lemma 9, then
ZAk
�c�kz + 1�
1
�� ��k(dz) <1
and Z t
0
ZAk
�c�kz + 1�
1
�� ��k(dz)ds <1:
Proof. We prove this lemma for Ak = (�1; 0) since the other two cases can be dealt with
similarly. As c�k � 1� 1 , it su¢ ces to show
Z �1=2
�1
�c�kz + 1�
1
�� ��k(dz) <1 (33)
and Z t
0
Z �1=2
�1
�c�kz + 1�
1
�� ��k(dz)ds <1: (34)
42
Let
F (c) = �c�qk
��k+1
�1� 1
� �1 ZAk
�cz + 1� 1
�1� ��k(dz):
Then
F 0(c) = � �qk
��k��1� 1
� ZAk
z
�cz + 1� 1
�� ��k(dz)
F 00(c) =
�1� 1
� ZAk
z2�cz + 1� 1
�� �1��k(dz) > 0
implying that F (c) is a convex function.
We now consider two cases: c�k < 1 � 1 and c�k = 1 � 1
. If c�k < 1 � 1
, then c�k satis�es
the �rst-order condition as follows
F 0(c�k) = ��qk��k��1� 1
� ZAk
z
�c�kz + 1�
1
�� ��k(dz) = 0
implying
Z �1=2
�1
�c�kz + 1�
1
�� ��k(dz)
� �2Z �1=2
�1z
�c�kz + 1�
1
�� ��k(dz)
� �2Z 0
�1z
�c�kz + 1�
1
�� ��k(dz) =
2�qk��k
�1� 1
�� <1 (35)
yielding (33). If c�k = 1� 1 , then c�k satis�es the following condition:
F 0(c�k) = ��qk��k��1� 1
� ZAk
z
�c�kz + 1�
1
�� ��k(dz) � 0:
43
Hence, like (35), we can show
Z �1=2
�1
�c�kz + 1�
1
�� ��k(dz) �
2�qk��k
�1� 1
�� <1 (36)
�nishing the proof of (33).
Combining (35) and (36) gives
Z t
0
Z �1=2
�1
�c�kz + 1�
1
�� ��k(dz) <1
�nishing the proof of (34).
B.2 Proof of (18)
We now turn to the calculation of sup�Q� in (18) of Proposition 3. For simplicity, we let t = 0
in the proof. In this case, as mentioned in Bardhan and Chao (1996), the set � comprises
the triples � = (v; �; �), with � and � being strictly positive, satisfying (14) or equivalently,
v(t) = e�b (37)ZAk
�k(t)�k(t; z)Sgn(z)��
k(dz) = � �qk
��k
and ZE
�k(t; z)��k(dz) = 1
for t � 0 and k = 1; :::; n�d. We let �d denote the family of the triples � = f(v(t); �(t); �(t))gt2[0;T ]
solving the following optimization problem:
44
sup�2�d
ZAk
�(�k(t)�k(t; z))
1� 1 � 1 +
�1� 1
�(1� �k(t)�k(t; z))
���k(dz) (38)
subject to ZAk
�k(t)�k(t; z)Sgn(z)��
k(dz) = ��qk��k;
for k 2 f1; :::; n � dg. By letting x(z) = �k(s)�k(s; z), it is straightforward to conclude
that the optimization problem above is equivalent to the optimization problem (27) with the
constraint (28).
We prove the following lemma.
Lemma 11
sup�2�
E�he�(1�
1 )R T0 rds�
1 (��(T ))
1� 1
i= sup
�2�dE�he�(1�
1 )R T0 rds�
1 (��(T ))
1� 1
i;
where � = �(0; T ) = exp�R T
0H(�s)ds
�.
Proof. It su¢ ces to prove
sup�2�
E�he�(1�
1 )R T0 rds�
1 (��(T ))
1� 1
i� sup
�2�dE�he�(1�
1 )R T0 rds�
1 (��(T ))
1� 1
i: (39)
Let Nk(t; T ) denote the number of k-th type of jump in the interval (t; T ]. Note that for any
t 2 [0; T ],
��(T )1� 1
= (��(t))1� 1
(��(t; T ))1� 1
45
where
��(t; T ) = exp
��Z T
t
vT (s)dz(s)� 12
Z T
t
jjv(s)jj2ds�
�n�dYk=1
Nk(t;T )Yi=1
(�k(tki )�k(t
ki ; z
ki ))
� exp�Z T
t
ZAk
(1� �k(s)�k(s; z))��k�
�k(dz)ds
�:
Hence the optimal v�(t) and #�k(t) �k(t; z) only depend on the state variables Xt. Thus, if
we let �X denote the family of � with v(t) and �k(t)�k(t; z) only depending on the state
variables Xt, then
sup�2�
E�he�(1�
1 )R T0 rds�
1 (��(T ))
1� 1
i= sup
�2�XE�he�(1�
1 )R T0 rds�
1 (��(T ))
1� 1
i:
Hence, to prove (39), it su¢ ces to show the following result:
sup�2�X
E�he�(1�
1 )R T0 rds�
1 (��(T ))
1� 1
i� sup
�2�dE�he�(1�
1 )R T0 rds�
1 (��(T ))
1� 1
i: (40)
De�ne
e��(t) =
n�dYk=1
Nk(t)Yi=1
(�k(tki )�k(t
ki ; z
ki ))
1� 1
� exp�Z t
0
ZAk
(1� (�k(s)�k(s; z))1� 1
)��k�
�k(dz)ds
�:
46
Note that e��(t) can be rewritten ase��(t) =
n�dYk=1
Nk(t)Yi=1
e�k(tki )e�k(tki ; zki )� exp
�Z t
0
ZAk
(1� e�k(s)e�k(s; z))��k��k(dz)ds�
where
e�k(s) = (�k(s))1� 1
ZEk
(�k(s; z))1� 1
��k(dz)
e�k(s; z) =(�k(s; z))
1� 1 Z
Ak
(�k(s; z))1� 1
��k(dz)
(41)
By Jensen�s inequality,
ZAk
(�k(s; z))1� 1
��k(dz) �
�ZAk
�k(s; z)�k(dz)
�1� 1
= 1:
It is not di¢ cult to see
Z T
0
e�k(s)��kds �Z T
0
(�k(s))1� 1
��kds
�Z T
0
��1� 1
��k(s) + 1
���kds <1:
Thus, for � 2 �locX , e��(t) is a local martingale from C4 in Bremaud (1981). And moreover,
noticing that the process e��(t) is non-negative and the state variables Xt do not include
jumps, we have
E�he��(T )jFX
T
i� 1; (42)
47
where FXT is the �-algebra generated by fXt; 0 � t � Tg. De�ne
fk(�k�k; t) =
ZAk
�(�k(t)�k(t; z))
1� 1 � 1 +
�1� 1
�(1� �k(t)�k(t; z))
���k�
�k(dz):
Note that
�1 (��(T ))
1� 1 = �
1 ��b�(T )
�1� 1 e��(T ) exp
n�dXk=1
Z T
0
fk(�k�k; t)dt
!
hence, by (42), for � 2 �X ,
E�he�(1�
1 )R T0 rds�
1 (��(T ))
1� 1
i= E�
hE�he�(1�
1 )R T0 rds�
1 (��(T ))
1� 1 jFX
T
ii= E�
"e�(1�
1 )R T0 rds�
1 ��b�(T )
�1� 1 E�
he��(T )jFXT
iexp
n�dXk=1
Z T
0
fk(�k�k; t)dt
!#
� E�
"e�(1�
1 )R T0 rds�
1 ��b�(T )
�1� 1 exp
n�dXk=1
Z T
0
fk(�k�k; t)dt
!#: (43)
Let ��k(s)��k(s; z); k = 1; ::n� d; denote the optimal solution to the problem (38). By (31) in
the proof of Lemma 8,
��k(s)��k(s; z) =
�c�kz + 1�
1
�� �1� 1
�
and by Lemma 10,
ZAk
e��k(s; z)��k(dz) = 1Z t
0
e��k(s)��kds < 1;
48
where e��k(s) and e��k(s; z) are de�ned according to (41).As in Bremaud (1981), we de�ne the following stopping time
Tn =
8>><>>:infntje���(t�) +Xn�d
k=1
R t0e��k(s)�kds � n
o;
or 1:
According to C4 in Bremaud (1981), e���(t ^ Tn) is a martingale, and, like (42),E�he���(T ^ Tn)jFX
T
i= 1:
Therefore, by (43),
E�he�(1�
1 )R T0 rds�
1 (��(T ))
1� 1
i� E�
"e�(1�
1 )R T0 rds�
1 ��b�(T )
�1� 1 E� [e���(T ^ Tn)jFX
T ] exp
n�dXk=1
Z T
0
fk(��k�
�k; t)dt
!#
= E�
"e�(1�
1 )R T0 rds�
1 ��b�(T )
�1� 1 e���(T ^ Tn) exp
n�dXk=1
Z T
0
fk(��k�
�k; t)dt
!#
� E�
"e�(1�
1 )R T0 rds�
1 ��b�(T )
�1� 1 e���(T ^ Tn) exp
n�dXk=1
Z T^Tn
0
fk(��k�
�k; t)dt
!#
= E�
�e�(1�
1 )R T0 rds�
1 ��b�(T )
�1� 1
�e�q��(T ^ Tn)�1� 1
�(44)
with the second inequality following from the result
fk(��k�
�k; T ) � fk(�
�k�
�k; T ^ Tn)
because
x1�1 � 1 +
�1� 1
�(1� x) � 0; for x � 0:
49
We now prove
supnE
(�e�(1�
1 )R T0 rds�
1
�e�q��(T ^ Tn)�b�(T )�1� 1
� �1)<1:
By the de�nition of � given in Lemma 11, � � 1 since H(�s) � 0. Note that the non-negative
process e�q��(t ^ Tn)�b�(t) is a local martingale and thus a supermartingale. As a result,supnE�
(�e�(1�
1 )R T0 rds�
1
�e�q��(T ^ Tn)�b�(T )�1� 1
� �1)
� supnE�nhe�q��(T ^ Tn)�b�(T )io � 1 <1
implying the sequence �1
�e�q��(T ^ Tn)�b�(T )�1� 1 is uniformly integrable since
�1 > 1, and
therefore, by (44),
E�he�(1�
1 )R T0 rds�
1 (��(T ))
1� 1
i� lim
n!1E�
�e�(1�
1 )R T0 rds�
1 ��b�(T )
�1� 1
�e�q��(T ^ Tn)�1� 1
�= E�
�e�(1�
1 )R T0 rds�
1 ��b�(T )
�1� 1
�e�q��(T )�1� 1
�= E�
he�(1�
1 )R T0 rds�
1 (���(T ))
1� 1
i
for each � 2 �X . Here we have used the fact that �b�(T ) = �b��(T ) since v(t) = e�b by (37).Hence (40) is proved and this completes the proof of the lemma.
We now turn to sup� in (18) by solving the following optimization problem:
sup�E�he�(1�
1 )R T0 rds�
1 (��(T ))
1� 1
i:
50
In virtue of Lemma 11, we have
E�he�(1�
1 )R T0 rds�
1 (��(T ))
1� 1
i= E�
"e�(1�
1 )R T0 rds�
1 ��b�(T )
�1� 1 exp
n�dXk=1
Z T
0
fk(��k�
�k; t)dt
!#
= E
"e�(1�
1 )R T0 rds�(T )�
1 ��b�(T )
�1� 1 exp
n�dXk=1
Z T
0
fk(��k�
�k; t)dt
!#:
Note that E��(T )jFX
T
�= 1: Hence
sup�E�he�(1�
1 )R T0 rds�
1 (��(T ))
1� 1
i= sup
�E
"E
"e�(1�
1 )R T0 rds�(T )�
1 ��b�(T )
�1� 1 exp
n�dXk=1
Z T
0
fk(��k�
�k; t)dt
!�����FXT
##
= sup�E
"E��(T )jFX
T
�e�(1�
1 )R T0 rds�
1 ��b�(T )
�1� 1 exp
n�dXk=1
Z T
0
fk(��k�
�k; t)dt
!#
= sup�E
"e�(1�
1 )R T0 rds�
1 ��b�(T )
�1� 1 exp
n�dXk=1
Z T
0
fk(��k�
�k; t)dt
!#
= sup�E
�e�(1�
1 )R T0 rds
��b�(T )
�1� 1 exp
�Z T
0
g(�)dt
��;
where
g(�) =n�dXk=1
gk(�); (45)
gk(�) =(1� )�k 'k
ZAk
[#k k(t; z) ln(#k k(t; z)) + 1� #k k(t; z)]�k(dz)
+ fk(��k�
�k; t);
51
with #k > 0 andZAk
k(t; z)�k(dz) = 1: Note, from Lemma 9,
fk(��k�
�k; t) =
ZAk
[((��k(s)��k(s; z))
1� 1 � 1)
+
�1� 1
�(1� ��k(s)�
�k(s; z)]�k#k k(t; z)]�k(dz)
= infck2C�\D�
(�ck�qk +
ZAk
"1
�1� 1
� �1�ckz + 1�
1
�1� �1
��k#k k(t; z)�k(dz)
�= inf
ck2C�\D�
ZAk
(�ck�qk +
"1
�1� 1
� �1�ckz + 1�
1
�1� �1
��k#k k(t; z)
��k(dz)
Hence
gk(�) = infck2C�\D�
ZAk
hk(ck; #k k)�k(dz)
where
hk(c; #k k) =(1� )�k 'k
[#k k(t; z) ln(#k k(t; z)) + 1� #k k(t; z)]
�c�qk +1
"�1� 1
� �1�cz + 1� 1
�1� � 1#�k#k k(t; z):
For illustrative purposes, we �x #k = 1, that is, the investor is confortable with the
estimate of intensity of the kth jump but ambiguity averse to its size distribution. The
general case can be handled in a similar manner. For simplicity, we further assume that the
support Ak of the jump size is Ak = (�1;1). Then, by (32), C� \ D� = [0; 1 � 1= ). We
52
will prove the following
sup�E
�e�(1�
1 )R T0 rds
��b�(T )
�1� 1 exp
�Z T
0
g(�)dt
��= E
�e�(1�
1 )R T0 rds
��b�(T )
�1� 1 exp
�Z T
0
sup�g(�)dt
��; (46)
where
sup�g(�) = sup
g(�)
= sup
n�dXk=1
infck2C�\D�
ZAk
hk(ck; #k k)�k(dz)
=n�dXk=1
sup k
infck2C�\D�
ZAk
hk(ck; #k k)�k(dz)
=n�dXk=1
infck2C�\D�
sup k
ZAk
hk(ck; #k k)�k(dz)
subject toZAk
k(t; z)�k(dz) = 1; k = 1; :::; n�d: Here we make use of the Minimax Theorem
for the exchange infc2C�\D� and sup k , noting that the function is concave with respect to
k and convex with respect to c. As before, it su¢ ces to consider the each optimization
problem infc2C�\D� sup k separately.
As before, we now employ the Fenchel Duality Theorem and same notation again to solve
the above problem. To this end, de�ne
D = X =
�x :
ZAk
jx(z)j�k(dz) <1�
C =
�x 2 X :
ZAk
x(z)�k(dz) = 1
�f(x) = 0
�g(x) =
ZAk
g0(x(z))�k(dz):
53
where
g0(x) =
8>><>>:hk(c; x); 8x � 0;
�1; 8x < 0:
We �rst calculate the functional f � conjugate to f , given by
f �(x�) = supx2C[hx; x�i � f(x)] = sup
x2C
ZAk
x(z)x�(z)�k(dz)
where
hx; x�i =ZAk
x(z)x�(z)�k(dz); x 2 X and x� 2 X�:
The following result can be derived in the exactly same way as Lemma 7 and hence its
proof is omitted to save the space.
Lemma 12 The Conjugate Space eC� of f �(x�) is given byeC� = fx� : f �(x�) <1g = fec : ec 2 Rg
and
f �(x�) = supx2C[hx; x�i] = ec; for x� = ec 2 eC�:
We now turn to the calculation of the functional �g� conjugate to �g. According to the
de�nition, for x� 2 X�,
�g�(x�) = infx2D[hx; x�i � �g(x)]
= infx2D
ZAk
[x(z)x�(z)� g0(x(z))]�k(dz)
= infx2X
ZAk
[x(z)x�(z)� g0(x(z))]�k(dz):
54
The Conjugate Space of �g�(x�) is eD� = fx� : �g�(x�) > �1g. When using the Fenchel
Duality Theorem, we only need to calculate �g�(x�) for x� 2 eC�. To this end, we have thefollowing result.
Lemma 13 For x� = ec 2 C�,�g�(x�) = inf
x2X
ZAk
[x(z)x�(z)� g0(x(z))]�k(dz)
=
ZAk
infx2R[xx�(z)� g0(x)]�k(dz): (47)
Proof: Like Lemma 8, the inequality � is trivial, and it su¢ ces to prove �. To this end,
consider the minimization problem
infx2R[xec� g0(x)]:
Its �rst-order condition is given by
ec� 1
"�1� 1
� �1�cz + 1� 1
�1� � 1#�k �
(1� )�k 'k
lnx = 0:
Solving the above for x, we have
�k(z; c; ~c) = exp
( 'k
(1� )�k
"ec� 1
"�1� 1
� �1�cz + 1� 1
�1� � 1#�k
#); (48)
giving the probability of the worst case. And furthermore, it is not di¢ cult to verify that
for c < 1� 1 ; Z
Ak
�k(z; c; ~c)�k(dz) <1;
55
implying �k(z; c; ~c) 2 X and (47). And then the optimal value is
infx2R[xx�(z)� g0(x)] = �k(z; c; ~c)ec� g0(
�k(z; c; ~c))
= c�qk �(1� )�k 'k
(1� �k(z; c; ~c)):
As a result
supx2C\D
[�g(x)� f(x)]
= infec2 eC�k\ eD�k
[f �(x�)� �g�(x�)]
= infec2 eC�k\ eD�k
�ec� c�qk +(1� )�k 'k
ZAk
(1� �k(z; c; ~c))�k(dz)
�= infec2 eC�k\ eD�
k
ZAk
�ec� c�qk +(1� )�k 'k
(1� �k(z; c; ~c))
��k(dz)
and therefore
sup�g(�) =
n�dXk=1
infck2C�k\D�
k;eck2 eC�k\ eD�k
ZAk
gk(z; ck;eck)�k(dz);where
gk(z; c;ec) = ec� c�qk +(1� )�k 'k
(1� �k(z; c; ~c)):
C Proof of proposition 4
We now use the results obtained for CRRA utility function and the results in Bellini and
Frittelli (2002) to solve the optimal portfolio choice problem with a HARA utility function.
First we derive duality result for the model without ambiguity aversion and then we obtain
duality result for the model with ambiguity aversion by using the same idea as before. Note
56
that
U�U 0�1(y)
�� U(x) + y
�U 0�1(y)� x
�; 8x > 0; y > 0;
where U 0�1(y) = I(y) = y�1 +b: For simplicity, we consider t = 0 and let �t = exp
��R Ttr(s)ds
�.
Let Q denote the set of all equivalent martingale measures. Thus, for any � 2 Q and terminal
wealth WT , we have
U�U 0�1(y�0�T )
�� U(WT ) + y�0�T
�U 0�1(y�0�T )�WT
�;
and
E�U�U 0�1(y�0�T )
��� E [U(WT )] ; (49)
where y satis�es
E��0�TU
0�1(y�0�T )�= W0; (50)
giving
y =Eh(�0�T )
1� 1
i (W0 � bE(�0�T ))
:
We now prove that there exists a � 2 Q such that (�0�T )� 1 can be replicated and hence
I(y�0�T ) = y�1 (�0�T )
� 1 + b can be replicated. By Kramkov and Schachermayer (1999)
and by considering the utility function 11� x
1� , we have that there exists a � 2 Q such that
(�0�T )� 1 can be replicated. Furthermore, according to (49), we have
u(W0) = E�U(U 0�1(y�T ))
�=(W0 � bE(�0�T ))
1�
1� Eh(�0�T )
1� 1
i ; (51)
with y satisfying (50).
57
In the following, we use some results in Bellini and Frittelli (2002) to prove the following
u(W0) = inf&2Q
(W0 � bE(�0&T ))1�
1� Eh(�0&T )
1� 1
i :
To this end, we denote with L1 the space of essentially bounded random variables and de�ne
M0 = fW 2 L1 : E[�0&TW ] � W0 8& 2 Qg:
According to Lemma 1.1 and 1.2 of Bellini and Frittelli (2002) (note we do not need As-
sumption 1.3), we have
u(W0) = supW2M0
E[U(W )] (52)
By following (1.8) in Bellini and Frittelli (2002), we de�ne
U(W0; &; P ) = supW2M&
0
E[U(W )];
where M &0 = fW 2 L1 : E[�0&TW ] � W0g: It is easy to see from (52)
u(W0) = supW2M0
E[U(W )] � inf&2Q
U(W0; &; P ) (53)
since M0 v M &0 . As in Section 2.1 of Bellini and Frittelli (2002), we de�ne the concave
conjugate U�(x�) of the utility function U(x) as:
U�(x�) = infxfxx� � U(x)g:
58
In particular, for the HARA utility function U(x) in Section 3.2, we have
U�(x�) =
� 1(x�)1�
1 + bx�:
Hence, using Corollary 2.1 of Bellini and Frittelli (2002), we have
U(W0; &; P ) = min�2(0;1)
f�W0 � EP [U� (��0&T )]g
=(W0 � bE(�0&T ))
1�
1� Eh(�0&T )
1� 1
i
and hence, by (53),
u(W0) � inf&2Q
(W0 � bE(�0&T ))1�
1� Eh(�0&T )
1� 1
i :
From (51), we have
u(W0) = inf&2Q
(W0 � bE(�0&T ))1�
1� Eh(�0&T )
1� 1
i :
We now turn to the model with ambiguity aversion. By following the same approach as
that of Proposition 2, we can derive the indirect value function as
J(0;W0; X0) =1
1�
sup�sup�2Q�
(W0 � bE�(�0��(T )))1�
� E�
�exp
�1
Z T
0
(Hs + (1� )r)ds
���(T )
1� 1
�� :
59
From (16),
E� [�0��(T )] = E���0�
b�(T )�
q�(T )
�= E�
�E���0�
b�(T )�
q�(T )jFX
T
��= E�
��0�
b�(T )E
���q�(T )jFX
T
��= E�
��0�
b�(T )
�;
since E���q�(T )jFX
T
�= 1, implying that E� [�0��(T )] is independent of �: And therefore,
J(0;W0; X0) =(W0 � bE
��0�
b�(T )
�)1�
1�
� sup�sup�2Q�
E�
�exp
�1
Z T
0
(Hs + (1� )r)ds
���(T )
1� 1
�!
=(W0 � bE
��0�
b�(T )
�)1�
1� (f(0; X0))
;
where f(0; X0) is given by Proposition 3. Likewise, we can show
J(t;Wt; Xt) =(Wt � bE
��t�
b�(t; T )
�)1�
1� (f(t;Xt))
:
�
D Proof of proposition 5
Note that the optimal portfolio weight � and the corresponding indirect value function J of
the investor�s problem satisfy the HJB equation (6) with � subject to some no-bankruptcy
constraints on jump exposures. Speci�cally, if the k-th jump size has support on (0;1), then
the exposure to this jump satis�es ��qk � 0; if it has support on (�1; 0), then the exposure
60
satis�es ��qk � 1; if it has support on (�1;1), then the exposure satis�es 0 � ��qk � 1.
Hence, the set Fk in (19) is given by
Fk =
8>>>>>><>>>>>>:(0;1); 8Ak = (0;1);
(�1; 1); 8Ak = (�1; 0);
(0; 1); 8Ak = (�1;1):
Without loss of generality, we assume all jumps are positive because other cases can be
handled in exactly the same manner. From Proposition 4, J(t;W;X) = (W�b�t)1� 1� (f(X; t)) ,
plugging it into above equation (6) and taking the �rst order condition with respect to �
lead to
0 = (b� r1n)W (W � b�t)� (f(X; t)) � �b�
>b �
>W 2(W � b�t)� �1(f(X; t))
+ �b�t�x>W (W � b�t)
� (f(X; t)) �1fX
+W (f(X; t)) n�dXk=1
�k
ZAk
[W (1 + ��qkz)� b�t]� �qkz�k(dz) (54)
+n�dXk=1
yk�qk
where (y1; :::; yn�d) are called Lagrangian Multipliers satisfying the standard complimentary
slackness conditions
��qk > 0; yk = 0 or ��qk = 0; yk � 0
for k = 1; :::; n� d. By the de�nition of the market price of risk e� given by (11), we have�
�e�be�q�= �be�b + �qe�q = b� r1n+�q(� � �)
61
Hence, by using notation e�b and e�q, and noticing�qe�q = [�q1; :::;�q(n�d)](e�q1; :::;e�qn�d)> = n�dX
k=1
�qke�qk(54) can be rewritten as
0 = �bW (W � b�t)� (f(X; t))
�e�b � e�>b W (W � b�t)�1 + �t�
x>fXf
�+
n�dXk=1
�qkZk
= �
�W (W � b�t)
� (f(X; t)) �e�b � e�>b W (W � b�t)
�1 + �t�x> fX
f
�Z
�
where Z = (Z1; :::; Zn�d)> and
Zk = W (f(X; t)) �(W � b�t)
� (e�qk � �kak) + �k
ZAk
[W (1 + e�qkz)� b�t]� z�k(dz)
�+ yk:
Consequently, as � is invertible,
e�b � e�>b W (W � b�t)�1 + �t�
x>fXf= 0
Zk = W (f(X; t)) �(W � b�t)
� (e�qk � �kak) + �k
ZAk
[W (1 + e�qkz)� b�t]� z�k(dz)
�+yk = 0
(55)
for k = 1; :::; n � d. Furthermore, for each k, the �rst order condition (55) with constraint
implies that e��qk is the optimal solution to the problemmaxe�qk�0(f(X; t))
�e�qk(W � b�t)� W (e�qk � �kak) +
�k1�
ZAk
[W (1 + e�qkz)� b�t]1� �k(dz)
�
62
which is identical to the problem (19). Hence, the optimal e��b is given bye��>b =
W � b�tW
"e�b + �t�
x>fXf
#
and the optimal e��qk solves the problem (19) for k = 1; :::; n� d, completing the proof. �
E Proof of proposition 6
As in Jin and Zhang (2012), we can use HJB equation (6) and Proposition 1 to show that
e��q and �� solve the following problem:sup�q2F
infe� D(�q; e�) = "e�q(�q1 + �q2e�) + 1
1� e�Z
E
[(e�qz + 1)1� � 1]�(dz) + e�(ln(e�=�)� 1)�
#;
(56)
where the �rst two terms on the right hand side are related to (19) while the last term is
associated with the penalty function. Noticing e��q and �� be interior optimal solutions to theoptimization problem above, we have the �rst-order conditions for e��q and �� given by
�q1 + �q2�� + ��
Z 1
�1z(1 + e��qz)� �(dz) = 0; (57)
and
�q2e��q + 1
1�
Z 1
�1[(1 + e��qz)1� � 1]�(dz) + 1
'ln
���
�
�= 0: (58)
Rearranging (58) gives (20). For (21), we di¤erentiate (57) with respect to and get
� @e��q@
Z 1
�1z2(1 + e��qz)� �1�(dz)
�Z 1
�1z(1 + e��qz)� ln(1 + e��qz)�(dz) = 0;
63
implying (21). The negativity of@e��q@
follows from that z ln(1 + e��qz) > 0;8z > �1 since
e��q 2 [0; 1): Note that from (57),
�q2 +
Z 1
�1z(1 + e��qz)� �(dz) = � �q1�� : (59)
Di¤erentiating (57) with respect to ��, we have
�q2+
Z 1
�1z(1 + e��qz)� �(dz)
� ��@e��q@��
Z 1
�1z2(1 + e��qz)� �1�(dz) = 0; (60)
and therefore, combining (59) and the result above yields (22). We now turn to the proof of
(23). Di¤erentiating (57) and (58) with respect to ' and using (59), we have
�q1��@��
@'+ ��A
@e��q@'
= 0; (61)
and
1
��@��
@'= B +
'�q1��
@e��q@'
; (62)
Solving the two linear equations above gives (23). To prove@e��q@'
> 0, it su¢ ce to show B < 0
under the conditions given in Proposition 8. Let
f(x) =1
1�
Z 1
�1[1� (1 + xz)1� ]�(dz)� �q2x:
Noticing that f(x) is a strictly convex function on [0; 1], we haveB = f(e��q) < maxff(0); f(1)g.Furthermore, by assumption, the jump size is Y = exp(U) � 1 with U = N(�Y ; �
2Y ) and
64
hence it is straightforward to calculate
f(1) =1
1�
Z 1
�1[1� (1 + z)1� ]�(dz)� �q2
=1
1�
�1� exp
�(1� )�Y +
(1� )2�2Y2
��� �q2; (63)
which is negative by assumption. Thus, we have B = maxff(0); f(1)g < 0 since f(0) = 0:
From (61) and by noticing@e��q@'
> 0 and �q1 > 0, we have@��
@'< 0; completing the proof.
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