# Optimal Portfolios in Lévy Markets under State-Dependent Bounded Utility Functions

Authors: José E. Figueroa-López; Jin Ma
Journal: International Journal of Stochastic Analysis
 (2010)

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## Abstract

Motivated by the so-called shortfall risk minimization problem,
we consider Merton's portfolio optimization problem in a non-Markovian
market driven by a Lévy process, with a bounded state-dependent utility
function. Following the usualdual variational approach, we show that the
domain of the dual problem enjoys an explicit “parametrization,” built on
a multiplicative optional decomposition for nonnegative supermartingales
due to Föllmer and Kramkov (1997). As a key step we prove a closure property
for integrals with respect to a fixed Poisson random measure, extending a
result by Mémin (1980). In the case where either the Lévy measure ν of Z has finite number of atoms or ΔSt/St−=ζtϑ(ΔZt) for a process ζ and a
deterministic function ϑ, we characterize explicitly the admissible trading
strategies and show that the dual solution is a risk-neutral local martingale.


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## Body

## 1. Introduction

The task of determining good trading strategies is a fundamental problem in mathematical finance. A typical approach to this problem aims at finding the trading strategy that maximizes, for example, the final expected utility, which is defined as a concave and increasing functionU:ℝ→ℝ∪{-∞} of the final wealth. There are, however, many applications where a utility function varies with the underlying securities, or even random. For example, if the market is incomplete, it is often more beneficial to allow certain degree of “shortfall” in order to reduce the “super-hedging cost” (see, e.g., [1, 2] for more details). Mathematically, such a shortfall risk is often quantified by the expected loss(1.1)𝔼[L((H-VT)+)],
where L:ℝ+→ℝ is a convex increasing “loss” function, H is a contingent claim, and V is the value process that is subject to the constraint V0≤z, for a given initial endowment z≥0.The above shortfall minimizing problem can be easily recast as a utility maximization problem with abounded state-dependent utility of the form(1.2)U(v;ω):=L(H(ω))-L((H(ω)-v)+),ω∈Ω,
as it was first pointed out by Föllmer and Leukert [3] (see Definition 2.3 for a formal description of the general bounded state-dependent utility). Then, the minimal shortfall risk cost is given by(1.3)u(z):=sup{𝔼[U(VT(·),·)]:VisadmissibleandV0≤z},z≥0,
where the supremum is taken over all wealth processes {Vt}t≤T generated by admissible trading strategies. We should point out here that it is the boundedness and potential nondifferentiability of such utility function that give rise to some technical issues which make the problem interesting.The existence and essential uniqueness of the solution to the problem (1.3) in its various special forms have been studied by many authors see, for example, Cvitanić [4], Föllmer and Leukert [3], Xu [5], and Karatzas and Žitković [6], to mention a few. However, while the convex duality approach in [3] succeeds in dealing with the non-Markovian nature of the model, it does not seem to shed any light on how to compute, in a feasible manner, the optimal trading strategy, partly due to the generality of the model considered there. In this paper we will consider a specific but popular model driven by a Lévy process. Our goal is to narrow down the domain of dual problem so that the convex duality method holds true. Furthermore, we will try to give an explicit construction of the dual domain that contains the dual optimizer. Although at this point our results are still rather general, and at a theoretical level, we believe that this is a necessary step towards a feasible computational implementation of the convex duality method.While the utility maximization problem of this kind can be traced back to Merton [7, 8], in this paper we shall follow the convex duality method, suggested by Karatzas et al. [9], and later extended by Kunita [10] to general Lévy market models. However, we note that in [9] the utility function was required to be unbounded, strictly increasing and concave, continuously differentiable, and other technical assumptions including the so-called Inada conditions. On the other hand, since one of the key tools in [10] is an exponential representation for positive local supermartingales (see, e.g., [11, Lemma 4.2]), it is required that the utility function satisfies the same conditions as in [9] (in particular, unboundedness), plus that the dual domain Γ contains all positive “risk-neutral” local supermartingales. The boundedness and potential nondifferentiability of the utility function in our case thus cause some technical subtleties. For example, the dual optimal process can be 0 with positive probability, thus the representation theorem of Kunita [11, Lemma 4.2] does not apply anymore.A key element that we use to overcome these technical difficulties is an exponential representation theorem for nonnegative supermartingales by Föllmer and Kramkov [12]. This result leads to an explicit construction of the dual domain, based on those nonnegative supermartingales that can be written as stochastic exponentials ξ=ξ0ℰ(X-A), with A being an increasing process and X belonging to a class of semimartingales 𝒮 that is closed under Émery's topology. To validate this approach we prove a closure property for integrals with respect to a fixed compensated Poisson random measures, a result of interest on its own, which extends the analog property for integrals with respect to a fixed semimartingale due to Mémin [13]. Finally, unlike some previous works on the subject (see, e.g., Föllmer and Leukert [3] and Xu [5]), we do not use the so-called bipolar theorem of Kramkov and Schachermayer [14] to argue the attainability of the optimal final wealth. Instead, we shall rely on the fundamental characterization of contingent claims that are super replicable [1, 2], reducing the problem of finding the optimal primal solution to a super-eplication problem.We believe that the dual problem proposed in this paper offers several advantages. For example, since the dual class enjoys a fairly “explicit” description and “parametrization,” our results could be considered as a first step towards a feasible computational implementation of the covex duality method. Furthermore, the specific results we obtained for the Lévy market can be used to characterize the elements of the dual domain and the admissible trading strategies. In particular, if either (i) the jumps of the price processS are driven by the superposition of finitely many shot-noise Poisson processes, or (ii) ΔSt/St-=ζtϑ(ΔZt) for a process ζ and a deterministic function ϑ, we show that the dual solution is a risk-neutral local martingale.We would like to remark that some of our results are related to those in Xu [5], but there are essential differences. For example, the model in [5] exhibits only finite-jump activity and allows only downward price jumps (in fact, this assumption seems to be important for the approach there), while our model allows for general jump dynamics, and our approach is also valid for general additive processes, including the time-inhomogeneous cases considered in [5] (see (ii) of Section 6).The rest of the paper is organized as follows. In Section2 we introduce the financial model, along with some basic terminology that will be used throughout the paper. The convex duality method is revised in Section 3, where a potential optimal final wealth is constructed. An explicit description of a dual class and a characterizations of the dual optimum and admissible trading strategies are presented in Section 4. In Section 5 we show that the potential optimal final wealth is attained by an admissible trading strategy, completing the proof of the existence of optimal portfolio. In Section 6 we give some concluding remarks. Some necessary fundamental theoretical results, such as the exponential representation for nonnegative supermartingales of Föllmer and Kramkov [12] and the closure property for integrals with respect to Poisson random measures, are collected in Appendix A. Finally, Appendix B briefly outlines the proofs of the convex duality results used in the paper.


## 2. Notation and Problem Formulation

Throughout this paper we assume that all the randomness comes from a complete probability space(Ω,ℱ,ℙ), on which there is defined a Lévy process Z with Lévy triplet (σ2,ν,0) (see Sato [15] for the terminology). By the Lévy-Itô decomposition, there exist a standard Brownian motion W and an independent Poisson random measure N on ℝ+×ℝ∖{0} with mean measure 𝔼N(dt,dz)=ν(dz)dt, such that(2.1)Zt=σWt+∫0t∫|z|≤1zÑ(ds,dz)+∫0t∫|z|>1zN(ds,dz),
where Ñ(dt,dz):=N(dt,dz)-ν(dz)dt. Let 𝔽:={ℱt}t≥0 be the natural filtration generated by W and N, augmented by all the null sets in ℱ so that it satisfies the usual conditions (see, e.g., [16]).


### 2.1. The Market Model

We assume that there are two assets in the market: a risk free bond (or money market account), and a risky asset, say, a stock. The case of multiple stocks, such as the one studied in [10], can be treated in a similar way without substantial difficulties (see Section 6 for more details). As it is customary all the processes are taken to be discounted to the present value so that the value Bt of the risk-free asset can be assumed to be identically equal to 1. The (discounted) price of the stock follows the stochastic differential equation(2.2)dSt=St-{btdt+σtdWt+∫ℝ0v(t,z)Ñ(dt,dz)},
where ℝ0:=ℝ∖{0}, b∈Lloc1, σ∈Lloc2(W), and v∈Gloc(N) (see [17] for the terminology). More precisely, b, σ, and v are predictable processes such that v(·,·)>-1 a.s. (hence, S·>0 a.s.), and that the processes(2.3){∫0t|bs|ds}t≥0,{∫0t|σs|2ds}t≥0,{(∑u≤tv2(u,ΔZu))1/2}t≥0
are locally integrable with respect to time. Even though we will work with a finite horizon [0,T] later on, we choose to define our market model on ℝ+. Finally, we assume that the market is free of arbitrage so that there exists at least one risk-neutral probability measure ℚ such that the (discounted) process St, 0≤t≤T, is an 𝔽-local martingale under ℚ. Throughout, ℳ will stand for the class of all equivalent risk neutral measures ℚ. It is relevant to mention that we do not impose market completeness, and hence, the class ℳ is not assumed to be a singleton.


### 2.2. Admissible Trading Strategies and the Utility Maximization Problem

A trading strategy is determined by a predictable locally bounded processβ representing the proportion of total wealth invested in the stock. Then, the resulting wealth process is governed by the stochastic differential equation(2.4)Vt=w+∫0tVs-βsSs-dSs,0<t≤T,
where w stands for the initial endowment. For future reference, we give a precise definition of “admissible strategies.”Definition 2.1.
The processVw,β:=V solving (2.4) is called the value process corresponding to the self-financing portfolio with initial endowment w and trading strategy β. We say that a value process Vw,β is “admissible” or that the process β is “admissible” for w if Vtw,β≥0,forallt∈[0,T].For a given initial endowmentw, we denote the set all admissible strategies for w by 𝒰adw, and the set of all admissible value processes by 𝒱adw. In light of the Doléans-Dade stochastic exponential of semimartingales (see, e.g., [17, Section I.4f]), one can easily obtain necessary and sufficient conditions for admissibility.Proposition 2.2.
A predictable locally bounded processβ is admissible if and only if
(2.5)ℙ[{ω∈Ω:βtv(t,ΔZt)≥-1,fora.e.t≤T}]=1.To define our utility maximization problem, we begin by introducing thebounded state-dependent utility function.Definition 2.3.
A random functionU:ℝ+×Ω↦ℝ+ is called a “bounded and state-dependent utility function” if (1)
U(·,ω) is nonnegative, nondecreasing, and continuous on [0,∞);(2)
for each fixedw, the mapping ω↦U(w,ω) is ℱT-measurable;(3)
there is anℱT-measurable, positive random variable H such that for all ω∈Ω, U(·,ω) is a strictly concave differentiable function on (0,H(ω)), and it holds that(2.6)U(w,ω)≡U(w⋀H(ω),ω),w∈ℝ+,(2.7)𝔼[U(H;·)]<∞.Notice that theℱT-measurability of the random variable ω→U(VT(ω),ω) is automatic because U(w,ω) is ℬ([0,∞))×ℱT-measurable in light of the above conditions (1) and (2). We remark that while assumption (2.7) is merely technical, assumption (2.6) is motivated by the shortfall risk measure (1.2). Our utility optimization problem is thus defined as(2.8)u(z):=sup{𝔼[U(VT(·),·)]:V∈𝒱adwwithw≤z}
for any z>0. We should note that the above problem is relevant only for those initial wealths z that are smaller than w¯:=supℚ∈ℳ𝔼ℚ{H}, the super-hedging cost of H. Indeed, if z≥w¯, then there exists an admissible trading strategy β* for z such that VTz,β*≥H almost surely, and consequently, u(z)=𝔼[U(H,·)] (see [1, 2] for this super-hedging result).Our main objectives in the rest of the paper are the following: (1) Define the dual problem and identify the relation between the value functions of the primal and the dual problems; (2) By suitably defining the dual domain, prove the attainability of the associated dual problem; (3) Show that the potential optimum final wealth induced by the minimizer of the dual problem can be realized by an admissible portfolio. We shall carry out these tasks in the remaining sections.


## 2.1. The Market Model

We assume that there are two assets in the market: a risk free bond (or money market account), and a risky asset, say, a stock. The case of multiple stocks, such as the one studied in [10], can be treated in a similar way without substantial difficulties (see Section 6 for more details). As it is customary all the processes are taken to be discounted to the present value so that the value Bt of the risk-free asset can be assumed to be identically equal to 1. The (discounted) price of the stock follows the stochastic differential equation(2.2)dSt=St-{btdt+σtdWt+∫ℝ0v(t,z)Ñ(dt,dz)},
where ℝ0:=ℝ∖{0}, b∈Lloc1, σ∈Lloc2(W), and v∈Gloc(N) (see [17] for the terminology). More precisely, b, σ, and v are predictable processes such that v(·,·)>-1 a.s. (hence, S·>0 a.s.), and that the processes(2.3){∫0t|bs|ds}t≥0,{∫0t|σs|2ds}t≥0,{(∑u≤tv2(u,ΔZu))1/2}t≥0
are locally integrable with respect to time. Even though we will work with a finite horizon [0,T] later on, we choose to define our market model on ℝ+. Finally, we assume that the market is free of arbitrage so that there exists at least one risk-neutral probability measure ℚ such that the (discounted) process St, 0≤t≤T, is an 𝔽-local martingale under ℚ. Throughout, ℳ will stand for the class of all equivalent risk neutral measures ℚ. It is relevant to mention that we do not impose market completeness, and hence, the class ℳ is not assumed to be a singleton.


## 2.2. Admissible Trading Strategies and the Utility Maximization Problem

A trading strategy is determined by a predictable locally bounded processβ representing the proportion of total wealth invested in the stock. Then, the resulting wealth process is governed by the stochastic differential equation(2.4)Vt=w+∫0tVs-βsSs-dSs,0<t≤T,
where w stands for the initial endowment. For future reference, we give a precise definition of “admissible strategies.”Definition 2.1.
The processVw,β:=V solving (2.4) is called the value process corresponding to the self-financing portfolio with initial endowment w and trading strategy β. We say that a value process Vw,β is “admissible” or that the process β is “admissible” for w if Vtw,β≥0,forallt∈[0,T].For a given initial endowmentw, we denote the set all admissible strategies for w by 𝒰adw, and the set of all admissible value processes by 𝒱adw. In light of the Doléans-Dade stochastic exponential of semimartingales (see, e.g., [17, Section I.4f]), one can easily obtain necessary and sufficient conditions for admissibility.Proposition 2.2.
A predictable locally bounded processβ is admissible if and only if
(2.5)ℙ[{ω∈Ω:βtv(t,ΔZt)≥-1,fora.e.t≤T}]=1.To define our utility maximization problem, we begin by introducing thebounded state-dependent utility function.Definition 2.3.
A random functionU:ℝ+×Ω↦ℝ+ is called a “bounded and state-dependent utility function” if (1)
U(·,ω) is nonnegative, nondecreasing, and continuous on [0,∞);(2)
for each fixedw, the mapping ω↦U(w,ω) is ℱT-measurable;(3)
there is anℱT-measurable, positive random variable H such that for all ω∈Ω, U(·,ω) is a strictly concave differentiable function on (0,H(ω)), and it holds that(2.6)U(w,ω)≡U(w⋀H(ω),ω),w∈ℝ+,(2.7)𝔼[U(H;·)]<∞.Notice that theℱT-measurability of the random variable ω→U(VT(ω),ω) is automatic because U(w,ω) is ℬ([0,∞))×ℱT-measurable in light of the above conditions (1) and (2). We remark that while assumption (2.7) is merely technical, assumption (2.6) is motivated by the shortfall risk measure (1.2). Our utility optimization problem is thus defined as(2.8)u(z):=sup{𝔼[U(VT(·),·)]:V∈𝒱adwwithw≤z}
for any z>0. We should note that the above problem is relevant only for those initial wealths z that are smaller than w¯:=supℚ∈ℳ𝔼ℚ{H}, the super-hedging cost of H. Indeed, if z≥w¯, then there exists an admissible trading strategy β* for z such that VTz,β*≥H almost surely, and consequently, u(z)=𝔼[U(H,·)] (see [1, 2] for this super-hedging result).Our main objectives in the rest of the paper are the following: (1) Define the dual problem and identify the relation between the value functions of the primal and the dual problems; (2) By suitably defining the dual domain, prove the attainability of the associated dual problem; (3) Show that the potential optimum final wealth induced by the minimizer of the dual problem can be realized by an admissible portfolio. We shall carry out these tasks in the remaining sections.


## 3. The Convex Duality Method and the Dual Problems

In this section we introduce thedual problems corresponding to the primal problem (1.3) and revise some standard results of convex duality that are needed in the sequel. Throughout, Ũ represents the convex dual function of U(·;ω), defined as(3.1)Ũ(y,ω):=sup0≤z≤H(ω){U(z,ω)-yz}.
We observe that the function Ũ is closely related to the Legendre-Fenchel transformation of the convex function -U(-z). It can be easily checked that Ũ(·;ω) is convex and differentiable everywhere, for each ω. Furthermore, if we denote the generalized inverse function of U′(·,ω) by(3.2)I(y,ω):=inf{z∈(0,H(ω))∣U′(z,ω)<y},
with the convention that inf∅=∞, then it holds that(3.3)Ũ′(y,ω)=-(I(y;ω)⋀H),∀y>0,
and the function Ũ has the following representation:(3.4)Ũ(y,ω)=U(I(y,ω)⋀H(ω),ω)-y(I(y,ω)⋀H(ω)).Remark 3.1.
We point out that the random fields defined in (3.1) and (3.2) are ℬ([0,∞))×ℱT-measurable. For instance, in the case of Ũ, we can write
(3.5)Ũ(y,ω)=supz≥0{U(z,ω)-yz}1{z≤H(ω)},
and we will only need to check that (y,ω)→{U(z,ω)-yz}1{z≤H(ω)} is jointly measurable for each fixed z. This last fact follows because the random field in question is continuous in the spatial variable y for each ω and is ℱT-measurable for each y. In light of (3.3), it transpires that the random field I(y,ω)⋀H(ω) is jointly measurable. Given that the subsequent dual problems and corresponding solutions are given in terms of the fields Ũ(y,ω) and I(y,ω)⋀H(ω) (see Definition 3.3 and Theorem 3.5), the measurability of several key random variable below is guaranteed.Next, we introduce the traditionaldual class (cf. [14]).Definition 3.2.
LetΓ̃ be the class of nonnegative supermartingales ξ such that (i)
0≤ξ(0)≤1,(ii)
for each locally bounded admissible trading strategyβ, {ξ(t)Vtβ}t≤T is a supermartingale.To motivate the construction of the dual problems below we note that ifξ∈Γ̃ and V is the value process of a self-financing admissible portfolio with initial endowment V0≤z, then 𝔼[ξ(T)(VT⋀H)]≤z, and it follows that(3.6)𝔼[U(VT,·)]≤𝔼[U(VT⋀H,·)]-y(𝔼[ξ(T)(VT⋀H)]-z)≤𝔼{sup0≤z′≤H(·){U(z′,·)-yξ(T)z′}}+zy=𝔼{Ũ(yξ(T),·)}+zy
for any y≥0. The dual problem is defined as follows.Definition 3.3.
Given a subclassΓ⊂Γ̃, the minimization problem
(3.7)vΓ(y):=infξ∈Γ𝔼[Ũ(yξ(T),ω)],y>0,
is called the “dual problem induced by Γ.” The class Γ is referred to as a dual domain (or class) and vΓ(·) is called its dual value function.Notice that, by (3.6) and (3.7), we have the following weak duality relation between the primal and dual value functions:(3.8)u(z)≤vΓ(y)+zy,
valid for all z,y≥0. The effectiveness of the dual problem depends on the attainability of the lower bound in (3.8) for some y*=y*(z)>0 (in which case, we say that strong duality holds), and the attainability of its corresponding dual problem (3.7). The following well-known properties will be needed for future reference. Their proofs are standard and are outlined in Appendix B for the sake of completeness.Proposition 3.4.
The dual value functionvΓ corresponding to a subclass Γ of Γ̃ satisfies the following properties (1)
vΓ is nonincreasing on (0,∞) and 𝔼[U(0;·)]≤vΓ(y)≤𝔼[U(H;·)].(2)
If(3.9)0<wΓ:=supξ∈Γ𝔼[ξ(T)H]<∞,
then vΓ is uniformly continuous on (0,∞), and
(3.10)limy↓0𝔼[U(H;·)]-vΓ(y)y=supξ∈Γ𝔼[ξ(T)H].(3)
There exists a processξ̃∈Γ̃ such that 𝔼[Ũ(yξ̃(T),·)]≤vΓ(y).(4)
IfΓ is a convex set, then (i)vΓ is convex, and with (ii) there exists a ξ*∈Γ̃ attaining the minimum vΓ(y). Furthermore, the optimum ξ* can be “approximated” by elements of Γ in the sense that there exists a sequence {ξn}n⊂Γ for which ξn(T)→ξ*(T), a.s.We now give a result that is crucial for proving the strong duality in (3.8). The result follows from arguments quite similar to those in [9, Theorem 9.3]. For the sake of completeness, we outline the proof in Appendix B.Theorem 3.5.
Suppose that (3.9) is satisfied and  Γ is convex, then, for any  z∈(0,wΓ), there exist  y(z)>0 and  ξy(z)*∈Γ̃ such that  (i)
𝔼[Ũ(y(z)ξy(z)*(T),ω)]≤𝔼[Ũ(y(z)ξ(T),ω)],forallξ∈Γ;(ii)
𝔼[VzΓξy(z)*(T)]=z, where
(3.11)VzΓ:=I(y(z)ξy(z)*(T))⋀H;(iii)
u(z)≤𝔼[U(VzΓ;ω)].We note that Theorem3.5 provides essentially an upper bound for the optimal final utility of the form 𝔼[U(VzΓ;ω)], for certain “reduced” contingent claim VzΓ≤H. By suitably choosing the dual class Γ, we shall prove in the next two sections that this reduced contingent claim is (super-) replicable with an initial endowment z.


## 4. Characterization of the Optimal Dual

We now give a full description of a dual classΓ for which strong duality, that is, u(z)=vΓ(y)+zy, holds. Denote 𝒱+ to be the class of all real-valued càdlàg, nondecreasing, adapted processes A null at zero. We will call such a process “increasing.” In what follows we let ℰ(X) be the Doléans-Dade stochastic exponential of the semimartingale X (see, e.g., [17] for their properties). Let(4.1)𝒮:={Xt:=∫0tG(s)dWs+∫0t∫ℝ0F(s,z)Ñ(ds,dz):F≥-1},
and consider the associated class of exponential local supermartingales:(4.2)Γ(𝒮):={ξ:=ξ0ℰ(X-A):X∈𝒮,Aincreasing,andξ≥0}.
In (4.1), we assume that G∈Lloc2(W), F∈Gloc(N), and that F(t,·)=G(t)=0, for all t≥T. The following result shows not only that the class(4.3)Γ:=Γ̃∩Γ(𝒮)
is convex, but also that the dual optimum, whose existence is guaranteed from Theorem 3.5, remains in Γ. The proof of this result is based on a powerful representation for nonnegative supermartingales due to Föllmer and Kramkov [12] (see Theorem A.1 in Appendix A), and a technical result about the closedness of the class of integrals with respect to Poisson random measures, under Émery’s topology. We shall defer the presentation of these two fundamental results to Appendix A in order to continue with our discussion of the dual problem.Theorem 4.1.
The classΓ is convex, and if (3.9) is satisfied, the dual optimum  ξy(z)* of Theorem 3.5 belongs to  Γ, for any  0<z<wΓ.Proof.
Let us check that𝒮 meets with the conditions in Theorem A.1. Indeed, each X in 𝒮 is locally bounded from below since, defining τn:=inf{t≥0:Xt<-n},(4.4)Xtτn≥Xτn--(ΔXτn)-1τn<∞≥-n-1,
where (x)-=-x1x<0. Condition (i) of Theorem A.1 is straightforward, while condition (ii) follows from Theorem A.3. Finally, condition (iii) holds because the processes in 𝒮 are already local martingales with respect to ℙ and hence ℙ∈𝒫(𝒮) with A𝒮(ℙ)≡0. By Proposition A.2 we conclude that Γ(𝒮) is convex and closed under Fatou convergence on dense countable sets. On the other hand, Γ̃ is also convex and closed under Fatou convergence, and thus so is the class Γ:=Γ̃∩Γ(𝒮). To check the second statement, recall that the existence of the dual minimizer ξy(z)* in Theorem 3.5 is guaranteed from Proposition 3.4, where it is seen that ξy(z)* is the Fatou limit of a sequence in Γ (see the proof of Proposition 3.4). This suffices to conclude that ξy(z)*∈Γ since Γ is closed under under Fatou convergence.In the rest of this section, we present some properties of the elements inΓ and of the dual optimum ξ*∈Γ. In particular, conditions on the “parameters” (G,F,A) so that ξ∈Γ(𝒮) is in Γ̃ are established. First, we note that without loss of generality, A can be assumed predictable.Lemma 4.2.
Let(4.5)ξ:=ξ0ℰ(X-A)∈Γ(𝒮).
Then, there exist a predictable process Ap∈𝒱+ and a process X̂∈𝒮 such that ξ=ξ0ℰ(X̂-Ap).Proof.
LetXt:=∫0tG(s)dWs+∫0t∫ℝ0F(s,z)Ñ(ds,dz)∈𝒮. Since F∈Gloc(N), there are stopping times τn′↗∞ such that
(4.6)𝔼∫0τn′∫ℝ|F(s,z)|1|F|>1ν(dz)ds<∞,
Compare with [17, Theorem II.1.33]. Now, define
(4.7)τn′′:=inf{t≥0:At>n},
and τn:=τn′⋀τn′′. Then,
(4.8)𝔼A∞τn=𝔼[Aτn-]+𝔼[ΔAτn]≤n+1+𝔼[|F(τn,Zτn)|]≤n+2+𝔼∫0τn∫ℝ|F(s,z)|1|F|>1ν(dz)ds<∞,
where we used that ΔXt-ΔAt≥-1. Therefore, A is locally integrable, increasing, and thus, its predictable compensator Ap exists. Now, by the representation theorem for local martingales (see [17, Theorem III.4.34]), the local martingale X′:=A-Ap admits the representation
(4.9)X·′:=∫0·G′(s)dWs+∫0·∫ℝ0F′(s,z)Ñ(ds,dz).
Finally, ξ=ξ0ℰ(X-A)=ξ0ℰ(X-X′-Ap). The conclusion of the proposition follows since X̂:=X-X′ is necessarily in 𝒮.The following result gives necessary conditions for a processξ∈Γ(𝒮) to belong to Γ̃. Recall that a predictable increasing process A can be uniquely decomposed as the sum of three predictable increasing processes,(4.10)A=Ac+As+Ad,
where Ac is the absolutely continuous part, As is the singular continuous part, and Atd=∑s≤tΔAs is the jump part (cf. [18, Theorem 19.61]).Proposition 4.3.
Letξ:=ξ0ℰ(X-A)≥0, where ξ0>0,
(4.11)Xt:=∫0tG(s)dWs+∫0t∫ℝ0F(s,z)Ñ(ds,dz)∈𝒮,
and A is an increasing predictable process. Let τ be the “sinking time” of the supermartingale ξ:
(4.12)τ:=supninf{t:ξt<1n}=inf{t:ΔXt=-1orΔAt=1}.
Also, let at=dAtc/dt. Then, {ξtSt}t≤T is a supermartingale if and only if the following two conditions are satisfied. (i)
There exist stopping timesτn↗τ such that
(4.13)𝔼∫0τn∫ℝ|v(s,z)F(s,z)|ν(dz)ds<∞.(ii)
Forℙ-a.e. ω∈Ω,
(4.14)ht≤at,for almost every t∈[0,τ(ω)], where
(4.15)ht:=bt+σtG(t)+∫ℝv(t,z)F(t,z)ν(dz).Proof.
Recall thatξ and S satisfy the SDE’s
(4.16)dξt=ξt-(G(t)dWt+∫ℝ0F(t,z)Ñ(dt,dz)-dAt),dSt=St-(btdt+σtdWt+∫ℝ0v(t,z)Ñ(dt,dz)).
Integration by parts and the predictability of A yield that
(4.17)ξtSt=localmartingale+∫0tbsξs-Ss-ds+∫0tσsG(s)ξs-Ss-ds-∫0tξs-Ss-dAs+∫0t∫ℝ0v(s,z)F(s,z)ξs-Ss-N(ds,dz).
Suppose that {ξtSt}t≥0 is a nonnegative supermartingale. Then, the integral ∫0t∫ℝv(s,z)F(s,z)ξs-Ss-N(ds,dz) must have locally integrable variation in light of the Doob-Meyer decomposition for supermartingale (see, e.g., [16, Theorem III.13]). Therefore, there exist stopping times τn1↗∞ such that
(4.18)𝔼∫0τn1∫ℝ|v(s,z)F(s,z)ξs-Ss-|ν(dz)ds<∞.
Then, (i) is satisfied with τn:=τn1⋀τn2⋀τn3, where τn2:=inf{t:ξt<1/n} and τn3:=inf{t:S̃t<1/n}. Next, we can write (4.17) as
(4.19)ξtSt=localmartingale-∫0tξs-Ss-(dAs-hsds).
By the Doob-Meyer representation for supermartingales and the uniqueness of the canonical decomposition for special semimartingales, the last integral must be increasing. Then, at≥ht for t≤τ since ξt->0 and ξt=0 for t≥τ (see [17, Theorem I.4.61]).
We now turn to the sufficiency of conditions (i) and (ii). Since{ξt-St-}t≥0 is locally bounded,
(4.20)∫0t∫ℝ|v(s,z)F(s,z)|ξs-Ss-1s≤τnν(dz)ds
is locally integrable. Then, from (4.17), we can write
(4.21)ξt⋀τnSt⋀τn=localmartingale-∫0tξs-Ss-1s≤τn(dAs-hsds).
Condition (ii) implies that {ξt⋀τnSt⋀τn} is a supermartingale, and by Fatou, {ξt⋀τSt⋀τ}t≥0 will be a supermartingale. This concludes the prove since ξt=0 for t≥τ, and thus, ξt⋀τSt⋀τ=ξtSt, for all t≥0.The following result gives sufficient and necessary conditions forξ∈Γ(𝒮) to belong to Γ̃. Its proof is similar to that of Proposition 4.3.Proposition 4.4.
Under the setting and notation of Proposition4.3, ξ∈Γ(𝒮) belongs to Γ̃ if and only if condition (i) in Proposition 4.3 holds and, for any locally bounded admissible trading strategies β,
(4.22)ℙ[{ω:htβt≤at,fora.e.t∈[0,τ(ω)]}]=1.The previous result can actually be made more explicit under additional information on the structure of the jumps exhibited by the stock price process. We consider two cases: when the jumps come from the superposition of shot-noise Poisson processes, and when the random fieldv exhibit a multiplicative structure. Let us first extend Proposition 2.2 in these two cases.Proposition 4.5.
(i) Suppose thatν is atomic with finitely many atoms {zi}i=1k, then, a predictable locally bounded strategy β is admissible if and only if ℙ×dt-a.e.
(4.23)-1maxiv(t,zi)∨0≤βt≤-1miniv(t,zi)⋀0.
(ii)
Suppose that v(t,z)=ζtϑ(z), for a predictable locally bounded process ζ such that ℙ×dt-a.e. ζt(ω)≠0 and ζt-1 is locally bounded, and a deterministic function ϑ such that ν({z:ϑ(z)=0})=0, then, a predictable locally bounded strategy β is admissible if and only if ℙ×dt-a.e.
(4.24)-1ϑ¯∨0≤βtζt≤-1ϑ̲⋀0,
where ϑ¯:=sup{ϑ(z):z∈supp(ν)} and ϑ̲:=inf{ϑ(z):z∈supp(ν)}.Proof.
From Proposition2.2, recall that ℙ-a.s.
(4.25)βtv(t,ΔZt)≥-1,
for a.e. t≤T. Then, for any closed set C⊂ℝ0, 0≤s<t, and A∈ℱs,
(4.26)∑s<u≤tχA(ω)χC(ΔZu){βuv(u,ΔZu)+1}≥0.
Taking expectation, we get
(4.27)𝔼∫stχA∫C{βuv(u,z)+1}ν(dz)du≥0.
Since such processes Hu(ω):=χA×(s,t](ω,u) generate the class of predictable processes, we conclude that ℙ×dt-a.e.
(4.28)-1≤βt∫Cv(t,z)ν(dz)ν(C).
Let us prove (ii) (the proof of (i) is similar). Notice that
(4.29)infz∈Uϑ(z)=infC⊂ℝ0∫Cϑ(z)ν(dz)ν(C)≤supC⊂ℝ0∫Cϑ(z)ν(dz)ν(C)=supz∈Uϑ(z),
where U is the support of ν. Suppose that infz∈Uϑ(z)<0<supz∈Uϑ(z). Then, by considering closed sets Cn,Cn′⊂ℝ0 such that
(4.30)∫Cnϑ(z)ν(dz)ν(Cn)↗supzϑ(z),∫Cn′ϑ(z)ν(dz)ν(Cn′)↘infzϑ(z),
as n→∞, we can prove the necessity. The other two cases (namely, infzϑ(z)≥0 or 0≥supzϑ(z)) are proved in a similar way. Sufficiency follows since, ℙ-a.s.,
(4.31){t≤T:βtζtv(ΔZt)<-1}⊂{t≤T:βtζtsupz∈Uv(z)<-1}∪{t≤T:βtζtinfz∈Uv(z)<-1}.Example 4.6.
It is worth pointing out some consequences(a)
In the time homogeneous case, wherev(t,z)=z, the extreme points of the support of ν (or what accounts to the same, the infimum and supremum of all possible jump sizes) determine completely the admissible strategies. For instance, if the Lévy process can exhibit arbitrarily large or arbitrarily close to -1 jump sizes, then
(4.32)0≤βt≤1;
a constraint that can be interpreted as absence of shortselling and bank borrowing (this fact was already pointed out by Hurd [19]).(b)
In the case thatϑ̲≥0, the admissibility condition takes the form -1/ϑ¯≤βtζt. If in addition ζ·<0 (such that the stock prices exhibit only downward sudden movements), then -1/(ϑ¯ζt)≥βt, and β·≡-c, with c>0 arbitrary, is admissible. In particular, from Proposition 4.4, if ξ∈Γ(𝒮) belongs to Γ̃, then a.s. htβt≤at,fora.e.t≤τ. This means that ξ∈Γ(𝒮)∩Γ̃ if and only if condition (i) in Proposition 4.3 holds and ℙ-a.s. ht≥0, for a.e. t≤τ. For a general ζ and still assuming that ϑ̲≥0, it follows that β is admissible and ξ∈Γ(𝒮)∩Γ̃ satisfy that ℙ-a.s.
(4.33)-1ϑ¯(ζt∨0)≤βt≤-1ϑ¯(ζt⋀0),htζt-11{t≤τ}≤0,
for a.e. t≥0.We now extend Proposition4.4 in the two cases introduced in Proposition 4.5. Its proof follows from Propositions 4.4 and 4.5.Proposition 4.7.
Suppose that either (i) or (ii) in Proposition4.5 is satisfied, in which case, define
(4.34)ĥt:={-htmaxiv(t,zi)∨01{ht<0}-htminiv(t,zi)⋀01{ht>0}if(i)holdstrue,-htζt-1ϑ¯∨01{htζt-1<0}-htζt-1ϑ̲⋀01{htζt-1>0}if(ii)holdstrue,
then, a process ξ∈Γ(𝒮) belongs to Γ̃ if and only if condition (i) in Proposition 4.3 holds, and for ℙ-a.e. ω, ĥt(ω)1{t≤τ(ω)}≤at(ω)1{t≤τ(ω)}, for a.e. t≥0.We remark that the casesϑ̲≥0 and ϑ¯≤0 do not lead to any absurd in the definition of ĥ above as we are using the convention that 0·∞=0. Indeed, for instance, if ϑ̲≥0, it was seeing that htζt-1≤0, for a.e. t≤τ, and thus, we set the second term in the definition of ĥ to be zero.Now we can give a more explicit characterization of the dual solutionξ*=ℰ(X*-A*) to problem (3.7), whose existence was established in Theorem 4.1. For instance, we will see that A* is absolutely continuous up to a predictable stopping time. Below, we refer to Proposition 4.3 for the notation.Proposition 4.8.
Letξ:=ξ0ℰ(X-A)∈Γ(𝒮), τA:=inf{t:ΔAt=1}, and Ãt:=∫0tasds+1{t≥τA}. The followings two statements hold true. (1)
ξ̃:=ξ0ℰ(X-Ã)≥ξ. Furthermore, ξ∈Γ̃ if and only if ξ̃∈Γ̃.(2)
Suppose that either of the two conditions in Proposition4.7 are satisfied and denote
(4.35)Ât:=∫0tĥs1s≤τds+1{t≥τA},
where ĥ is defined accordingly to the assumed case. Then, ξ·≤ξ̂·, and furthermore, the process ξ̂:=ξ0ℰ(X-Â) belongs to Γ̃ if ξ∈Γ̃.Proof.
LetAc,As,Ad denote the increasing predictable processes in decomposition (4.10) of A. Since A is predictable, there is no common jump times between X and A. Then,
(4.36)ξt=ξ0eXt-At-(1/2)〈Xc,Xc〉t∏s≤t(1+ΔXs)e-ΔXs∏s≤t(1-ΔAs)eΔAs≤ξ0eXt-Atc-(1/2)〈Xc,Xc〉t∏s≤t(1+ΔXs)e-ΔXs1{t<τA}=ξ̃t,
where we used that At-∑s≤tΔAs=Atc+Ats≥Atc, and ∏s≤t(1-ΔAs)≤1{t<τA}. Since both processes ξ and ξ̃ enjoy the same absolutely continuous part, and the same sinking time, the second statement in (1) is straightforward from Proposition 4.4. Part (2) follows from Proposition 4.7 since the process ât:=ĥt1t≤τ is nonnegative, predictable (since h is predictable), and locally integrable (since 0≤ĥ≤a).We remark that part (2) in Proposition4.8 remains true if we take Ât:=∫0tĥs1s≤τAds+1{t≥τA}. The following result is similar to Proposition 3.4 in Xu [5] and implies, in particular, that the optimum dual ξ* can be taken to be a local martingale.Proposition 4.9.
Suppose that either (i) or (ii) of Proposition4.5 is satisfied. Moreover, in the case of condition (ii), assume additionally that
(4.37)ν({z∈supp(ν)∖{0}:ϑ(z)=c})>0,
for c=ϑ̲ if ϑ¯>0, and for c=ϑ¯ if ϑ̲<0. Let ξ∈Γ̃∩Γ(𝒮). Then, there exists X̃∈𝒮 such that ξ̃:=ξ0ℰ(X̃)∈Γ̃ and ξ·≤ξ̃·. Furthermore, {ξ̃(t)Vtβ}t≤T is a local martingale for all locally bounded admissible trading strategies β.Proof.
Let us prove the case when condition (i) in Proposition4.5 is in force. In light of Proposition 4.8, we assume without loss of generality that At=∫0tatdt+1{t≥τA}, with at:=ĥt1{t≤τ}. Assume that miniv(t,zi)<0<maxiv(t,zi). Otherwise if, for instance, maxiv(t,zi)≤0, then it can be shown that ht≥0, a.s. (similarly to case (b) in Example 4.6), and the first term of ĥ is 0 under our convention that ∞·0=0. Notice that, in any case, one can find a predictable process z taking values on {zi}i=1n, such that
(4.38)ĥt=-htv(t,z(t)).
Write X̃·:=∫0·G(s)dWs+∫0·∫ℝ0F̃(s,z)dÑ(s,z) for an F̃∈Gloc(N) to be determined in the sequel. For ξ̃≥ξ it suffices to prove the existence of a field D satisfying both conditions below:
(4.39)(a)D≥0,(b)∫ℝ0D(t,z)ν(dz)1{t≤τ}≤ĥt1{t≤τ},
(then, F̃ is defined as D+F). Similarly, for ξ̂ to belong to Γ̃ it suffices that
(4.40)(c)ht+∫ℝ0v(t,z)D(t,z)ν(dz)=0.
Taking
(4.41)D(t,z):=-htv(t,z(t))ν({z(t)})1{z=z(t)},
clearly nonnegative, (b) and (c) hold with equality. Moreover, the fact that inequalities (c) hold with equality implies that {ξ̂(t)Vtβ}t≤T is a local martingale for all locally bounded admissible trading strategy β (this can be proved using the same arguments as in the sufficiency part of Proposition 4.3). Now suppose that condition (ii) in Proposition 4.5 holds. For simplicity, let us assume that ϑ̲<0<ϑ¯ (the other cases can be analyzed following arguments similar to Example 4.6). Notice that (4.37) implies the existence of a Borel C̲ (resp., C¯) such that ϑ(z)≡ϑ̲ on C̲ (resp., ϑ(z)≡ϑ¯ on C¯ ) and 0<ν(C̲),ν(C¯)<∞. Taking
(4.42)D(t,z):=-htζt-1ϑ¯ν(C¯)1C¯(z)1{htζt-1<0}-htζt-1ϑ̲ν(C̲)1C̲(z)1{htζt-1>0},(b) and (c) above will hold with equality.


## 5. Replicability of the Upper Bound

We now show that the tentative optimum final wealthVzΓ, suggested by the inequality (iii) in Theorem 3.5, is (super-) replicable. We will combine the dual optimality of ξ* with the super-hedging theorem, which states that given a contingent claim Ĥ satisfying w¯:=supℚ∈ℳ𝔼ℚ{Ĥ}<∞, one can find for any fixed z≥w¯ an admissible trading strategy β* (depending on z) such that VTz,β*≥Ĥ almost surely (see Kramkov [2], and also Delbaen and Schachermayer [1]). Recall that ℳ denotes the class of all equivalent risk neutral probability measures.Proposition 5.1.
Under the setting and conditions of Theorem3.5, for any 0<z<wΓ, there is an admissible trading strategy β* for z such that
(5.1)VTz,β*≥I(y(z)ξy(z)*(T))⋀H,
and thus, the optimum of u(z) is reached at the strategy β*. In particular,
(5.2)VTz,β*=I(y(z)ξy(z)*(T)),
when I(y(z)ξy(z)*(T))<H.Proof.
For simplicity, we writeξt*:=ξy(z)*(t), y=y(z), and
(5.3)V*=I(y(z)ξy(z)*(T))⋀H.
Fix an equivalent risk neutral probability measure ℚ∈ℳ, and let ξt′=(dℚ∣ℱt/dℙ∣ℱt) be its corresponding density processes. Here, ℚ|ℱt (resp., ℙ|ℱt) is the restriction of the measure ℚ (resp., ℙ) to the filtration ℱt. Under ℚ, S· is a local martingale, and then, for any locally bounded β, V·β is a ℚ-local martingale. By [17, Proposition III.3.8.c], ξ′Vβ is a ℙ-local martingale (necessarily nonnegative by admissibility), and thus, ξ′ is in Γ̃. On the other hand, ξ′ belongs to Γ(𝒮) due to the exponential representation for positive local martingales in Kunita [11] (alternatively, by invoking [17, Theorems III.8.3, I.4.34c, and III.4.34], ξ′∈Γ(𝒮) even if Z were just an additive process Z). By the convexity of the dual class Γ=Γ(𝒮)∩Γ̃ and the fact that ξ*∈Γ (see Theorem 4.1), ξ(ɛ):=ɛξ′+(1-ɛ)ξ* belongs to Γ, for any 0≤ɛ≤1. Moreover, since Ũ is convex and Ũ′(y)=-(I(y)⋀H),
(5.4)|Ũ(yξT(ɛ))-Ũ(yξT*)ε|≤yH|ξT′-ξT*|≤yH(ξT′+ξT*).
The random variable yH(ξT′+ξT*) is integrable since by assumption wΓ<∞. We can then apply dominated convergence theorem to get
(5.5)limɛ↓01ɛ{𝔼[Ũ(yξT(ɛ))]-𝔼[Ũ(yξT*)]}=-y𝔼[V*(ξT′-ξT*)],
which is nonnegative by condition (i) in Theorem 3.5. Then, using condition (ii) in Theorem 3.5,(5.6)𝔼ℚ[V*]=𝔼[V*ξT′]≤𝔼[V*ξT*]=z.
Since ℚ∈ℳ is arbitrary, supℚ∈ℳ𝔼ℚ[V*]≤z. By the super-hedging theorem, there is an admissible trading strategy β* for z such that
(5.7)VTz,β*≥I(y(z)ξy(z)*(T))⋀H.
The second statement of the theorem is straightforward since U(z) is strictly increasing on z<H.


## 6. Concluding Remarks

We conclude the paper with the following remarks.(i) The dual classΓ
The dual domain of the dual problem can be taken to be the more familiar class of equivalent risk-neutral probability measuresℳ. To be more precise, define(6.1)Γ¯:={ξt:=dℚ∣ℱtdℙ∣ℱt:ℚ∈ℳ}.
Since Γ¯ is obviously a convex subclass of Γ̃, Theorem 3.5 implies that, as far as(6.2)0<w¯:=supξ∈Γ¯𝔼[ξTH]<∞,
for each z∈(0,w¯), there exist y:=y(z)>0 and ξ*:=ξy(z)*∈Γ̃ (not necessarily belonging to Γ¯) such that (i)–(iii) in Theorem 3.5 hold with Γ=Γ¯. Finally, one can slightly modify the proof of Proposition 5.1, to conclude the replicability of(6.3)VzΓ¯:=I(yξT*)⋀H.
Indeed, in the notation of the proof of the Proposition 5.1, the only step which needs to be justified in more detail is that(6.4)𝔼[Ũ(yξT*)]≤𝔼[Ũ(yξT(ɛ))],
for all 0≤ɛ≤1, where ξ(ɛ)=ɛξ'+(1-ɛ)ξ* (here, ξ′ is a fixed element in Γ¯). The last inequality follows from the fact that, by Proposition 5.1 (c), ξ* can be approximated by elements {ξ(n)}n≥1 in Γ¯ in the sense that ξT(n)→ξT* a.s. Thus, ξ(ɛ) can be approximated by the elements ξ(ɛ,n):=ɛξ′+(1-ɛ)ξ(n) in Γ¯, for which we know that(6.5)𝔼[Ũ(yξT*)]≤𝔼[Ũ(yξT(ɛ,n))].
Passing to the limit as n→∞, we obtain (6.4).In particular we conclude that condition (6.2) is sufficient for both the existence of the solution to the primal problem and its characterization in terms of the dual solution ξ*∈Γ̃ of the dual problem induced by Γ=Γ¯. We now further know that ξ* belongs to the class Γ̃∩Γ(𝒮) defined in (4.3), and hence, enjoys an explicit parametrization of the form(6.6)ξ*:=ℰ(∫0·G*(s)dWs+∫0·∫ℝ0F*(s,z)Ñ(ds,dz)-∫0·as*ds),
for some triple (G*,F*,a*).(ii) Market driven by general additive models
Our analysis can be extended to more general multidimensional models driven by additive processes (i.e., processes with independent, possibly nonstationary increments; cf. Sato [15] and Kallenberg [20]). For instance, let (Ω,ℱ,ℙ) be a complete probability space on which is defined a ddimensional additive process Z with Lévy-Itô decomposition:(6.7)Zt=αt+ΣWt+∫0t∫{∥z∥>1}zN(ds,dz)+∫0t∫{∥z∥≤1}zÑ(ds,dz),
where W is a standard d-dimensional Brownian motion, N(dt,dz) is an independent Poisson random measure on ℝ+×ℝd, and Ñ(dt,dz)=N(dt,dz)-𝔼N(dt,dz). Consider a market model consisting of n+1 securities: one risk free bond with price(6.8)dBt:=rtBtdt,B0=1,t≥0,
and n risky assets with prices determined by the following stochastic differential equations with jumps:(6.9)dSti=St-i{btidt+∑j=1dσtijdWtj+∫ℝdvi(t,z)Ñ(ds,dz)},i=1,…,n,
where the processes r, b, σ, and v are predictable satisfying usual integrability conditions (cf. Kunita [10]). We assume that ℱ:=ℱ∞-, where 𝔽:={ℱt}t≥0 is the natural filtration generated by W and N; namely, ℱt:=σ(Ws,N([0,s]×A):s≤t,A∈ℬ(ℝd)). The crucial property, particular to this market model, that makes our analysis valid, is the representation theorem for local martingales relative to Z (see [17, Theorem III.4.34]). The definition of the dual class Γ given in Section 4 will remain unchanged, and only very minor details will change in the proof of Theorem A.3. Some of the properties of the results in Section 4 regarding the properties of Γ will also change slightly. We remark that, by taking a real (nonhomogeneous) Poisson process, the model and results of Chapter 3 in Xu [5] will be greatly extended. We do not pursue the details here due to the limitation of the length of this paper.(iii) Optimal wealth-consumption problem
Another classical portfolio optimization in the literature is that of optimal wealth-consumption strategies under a budget constraint. Namely, we allow the agent to spend money outside the market, while maintaining “solvency” throughout[0,T]. In that case the agent aims to maximize the cost functional that contains a “running cost”:(6.10)𝔼[U1(VT)+∫0TU2(t,ct)dt],
where c is the instantaneous rate of consumption. To be more precise, the cumulative consumption at time t is given by Ct:=∫0tcudu and the (discounted) wealth at time t is given by(6.11)Vt=w+∫0tβudSu-∫0tcudu.
Here, U1 is a (state-dependent) utility function and U2(t,·) is a utility function for each t. The dual problem can now be defined as follows:(6.12)vΓ(y)=infξ∈Γ𝔼[Ũ1(yξT)+∫0TŨ2(s,yξs)ds],
over a suitable class of supermartingales Γ. For instance, if the support of ν is [-1,∞), then Γ can be all supermartingales ξ such that 0≤ξ0≤1 and {ξtSt}t≤T is a supermartingale. The dual Theorem 3.5 can be extended for this problem. However, the existence of a wealth-consumption strategy pair (β,c) that attains the potential final wealth induced by the optimal dual solution (as in Section 5) requires further work. We hope to address this problem in a future publication.



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