Optimal Portfolios in L ´evy Markets under State-Dependent Bounded Utility Functions

Motivated by the so-called shortfall risk minimization problem, we consider Merton’s portfolio optimization problem in a non-Markovian market driven by a L´evy process, with a bounded state-dependent utility function. Following the usual dual 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 F¨ollmer and Kramkov (cid:2) 1997 (cid:3) . As a key step we prove a closure property for integrals with respect to a ﬁxed Poisson random measure, extending a result by M´emin (cid:2) 1980 (cid:3) . In the case where either the L´evy measure ν of Z has ﬁnite number of atoms or Δ S t /S t − (cid:5) ζ t ϑ (cid:2)Δ Z t (cid:3) 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.


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 function U : 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 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 V 0 ≤ z, for a given initial endowment z ≥ 0.

International Journal of Stochastic Analysis
The above shortfall minimizing problem can be easily recast as a utility maximization problem with a bounded state-dependent utility of the form 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 u z : sup{E U V T · , · : V is admissible and V 0 ≤ z}, z ≥ 0, 1.3 where the supremum is taken over all wealth processes {V t } 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 andŽitković 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 ξ 0 E X − A , with A being an increasing process and X belonging to a class of semimartingales S that is closed underÉmery'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 International Journal of Stochastic Analysis 3 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 process S are driven by the superposition of finitely many shot-noise Poisson processes, or ii ΔS t /S t − ζ t ϑ ΔZ t 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 Section 2 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.

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 σ 2 , ν, 0 see Sato 15

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 B t 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 and v ∈ G loc 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 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 S t , 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.

Admissible Trading Strategies and the Utility Maximization Problem
A trading strategy is determined by a predictable locally bounded process β representing the proportion of total wealth invested in the stock. Then, the resulting wealth process is governed by the stochastic differential equation where w stands for the initial endowment. For future reference, we give a precise definition of "admissible strategies." Definition 2.1. The process V w,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 V w,β is "admissible" or that the process β is "admissible" for w if V w,β t ≥ 0, for all t ∈ 0, T .
For a given initial endowment w, we denote the set all admissible strategies for w by U w ad , and the set of all admissible value processes by V w ad . In light of the Doléans-Dade International Journal of Stochastic Analysis 5 stochastic exponential of semimartingales see, e.g., 17, Section I.4f , one can easily obtain necessary and sufficient conditions for admissibility.
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 function v Γ corresponding to a subclass Γ of Γ satisfies the following properties then v Γ is uniformly continuous on 0, ∞ , and International Journal of Stochastic Analysis 3 There exists a process ξ ∈ Γ such that E U y ξ T , · ≤ v Γ y .
4 If Γ is a convex set, then i v Γ is convex, and with ii there exists a ξ * ∈ Γ attaining the minimum v Γ y . Furthermore, the optimum ξ * can be "approximated" by elements of Γ in the sense that there exists a sequence {ξ n } n ⊂ Γ for which ξ n 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, w Γ , there exist y z > 0 and ξ * y z ∈ Γ such that We note that Theorem 3.5 provides essentially an upper bound for the optimal final utility of the form E U V Γ z ; ω , for certain "reduced" contingent claim V Γ z ≤ 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.

Characterization of the Optimal Dual
We now give a full description of a dual class Γ for which strong duality, that is, u z v Γ 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 S : and consider the associated class of exponential local supermartingales: In 4.1 , we assume that G ∈ L 2 loc W , F ∈ G loc N , and that F t, · G t 0, for all t ≥ T . The following result shows not only that the class International Journal of Stochastic Analysis 9 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, underÉmery'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 class Γ is convex, and if 3.9 is satisfied, the dual optimum ξ * y z of Theorem 3.5 belongs to Γ, for any 0 < z < w Γ .
Proof. Let us check that S meets with the conditions in Theorem A.1. Indeed, each X in S is locally bounded from below since, defining τ n : inf{t ≥ 0 : X t < −n}, 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 A S 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, Γ is also convex and closed under Fatou convergence, and thus so is the class Γ : Γ ∩ Γ S . To check the second statement, recall that the existence of the dual minimizer ξ * y z in Theorem 3.5 is guaranteed from Proposition 3.4, where it is seen that ξ * y z is the Fatou limit of a sequence in Γ see the proof of Proposition 3.4 . This suffices to conclude that ξ * y z ∈ Γ since Γ is closed under under Fatou convergence.
In the rest of this section, we present some properties of the elements in Γ and of the dual optimum ξ * ∈ Γ. In particular, conditions on the "parameters" G, F, A so that ξ ∈ Γ S is in Γ are established. First, we note that without loss of generality, A can be assumed predictable.

Lemma 4.2. Let
Then, there exist a predictable process A p ∈ V and a process X ∈ S such that Compare with 17, Theorem II.1.33 . Now, define International Journal of Stochastic Analysis and τ n : τ n ∧ τ n . Then, where we used that ΔX t −ΔA t ≥ −1. Therefore, A is locally integrable, increasing, and thus, its predictable compensator A p exists. Now, by the representation theorem for local martingales see 17, Theorem III.4.34 , the local martingale X : A − A p admits the representation The conclusion of the proposition follows since X : X − X is necessarily in S.
The following result gives necessary conditions for a process ξ ∈ Γ S to belong to Γ. Recall that a predictable increasing process A can be uniquely decomposed as the sum of three predictable increasing processes, where A c is the absolutely continuous part, A s is the singular continuous part, and A d t s≤t ΔA s is the jump part cf. 18, Theorem 19.61 .
and A is an increasing predictable process. Let τ be the "sinking time" of the supermartingale ξ: Also, let a t dA c t /dt.
Proof. Recall that ξ and S satisfy the SDE's

4.16
Integration by parts and the predictability of A yield that

4.17
Suppose that {ξ t S t } t≥0 is a nonnegative supermartingale. Then, the integral t 0 R v s, z F s, z ξ s − S s − 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 τ 1 Then, i is satisfied with τ n : τ 1 n ∧ τ 2 n ∧ τ 3 n , where τ 2 n : inf{t : ξ t < 1/n} and τ 3 n : inf{t : S t < 1/n}. Next, we can write 4.17 as By the Doob-Meyer representation for supermartingales and the uniqueness of the canonical decomposition for special semimartingales, the last integral must be increasing. Then, a t ≥ h t for t ≤ τ since ξ t − > 0 and ξ t 0 for t ≥ τ see 17, Theorem I.4.61 .
We now turn to the sufficiency of conditions i and ii . Since International Journal of Stochastic Analysis is locally integrable. Then, from 4.17 , we can write Condition ii implies that {ξ t∧τ n S t∧τ n } is a supermartingale, and by Fatou, {ξ t∧τ S t∧τ } t≥0 will be a supermartingale. This concludes the prove since ξ t 0 for t ≥ τ, and thus, ξ t∧τ S t∧τ ξ t S t , for all t ≥ 0.
The following result gives sufficient and necessary conditions for ξ ∈ Γ S to belong to Γ. Its proof is similar to that of Proposition 4.3.
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 field v exhibit a multiplicative structure. Let us first extend Proposition 2.2 in these two cases.
ii Suppose that v t, z ζ t ϑ z , for a predictable locally bounded process ζ such that P × dta.e. ζ t ω / 0 and ζ −1 t 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.
Let us prove ii the proof of i is similar . Notice that where U is the support of ν. Suppose that inf z∈U ϑ z < 0 < sup z∈U ϑ z . Then, by considering closed sets C n , C n ⊂ R 0 such that as n → ∞, we can prove the necessity. The other two cases namely, inf z ϑ z ≥ 0 or 0 ≥ sup z ϑ z are proved in a similar way. Sufficiency follows since, P-a.s., 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 that ϑ ≥ 0, the admissibility condition takes the form −1/ϑ ≤ β t ζ t . If in addition ζ · < 0 such that the stock prices exhibit only downward sudden movements , then −1/ ϑζ t ≥ β t , and β · ≡ −c, with c > 0 arbitrary, is admissible. In particular, from Proposition 4.4, if ξ ∈ Γ S belongs to Γ, then a.s. h t β t ≤ a t , for a.e. t ≤ τ. This means that ξ ∈ Γ S ∩ Γ if and only if condition i in Proposition 4.3 holds and P−a.s. h t ≥ 0, for a.e. t ≤ τ. For a general ζ and still assuming that ϑ ≥ 0, it follows that β is admissible and ξ ∈ Γ S ∩ Γ satisfy that P-a.s.
We now extend Proposition 4.4 in the two cases introduced in Proposition 4.5. Its proof follows from Propositions 4.4 and 4.5. We remark that the cases ϑ ≥ 0 and ϑ ≤ 0 do not lead to any absurd in the definition of h above as we are using the convention that 0 · ∞ 0. Indeed, for instance, if ϑ ≥ 0, it was seeing that h t ζ −1 t ≤ 0, for a.e. t ≤ τ, and thus, we set the second term in the definition of h to be zero. Now we can give a more explicit characterization of the dual solution ξ * 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 4.7. Suppose that either (i) or (ii) in Proposition 4.5 is satisfied, in which case, define
2 Suppose that either of the two conditions in Proposition 4.7 are satisfied and denote where h is defined accordingly to the assumed case. Then, ξ · ≤ ξ · , and furthermore, the process ξ :

Proof. Let A c , A s , A d 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, Since both processes ξ and ξ 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 a t : h t 1 t≤τ is nonnegative, predictable since h is predictable , and locally integrable since 0 ≤ h ≤ a .
We remark that part 2 in Proposition 4.8 remains true if we take A t : 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 Proposition 4.5 is satisfied. Moreover, in the case of condition (ii), assume additionally that
for c ϑ if ϑ > 0, and for c ϑ if ϑ < 0. Let ξ ∈ Γ ∩ Γ S . Then, there exists X ∈ S such that ξ : ξ 0 E X ∈ Γ and ξ · ≤ ξ · . Furthermore, { ξ t V β t } t≤T is a local martingale for all locally bounded admissible trading strategies β.
Proof. Let us prove the case when condition i in Proposition 4.5 is in force. In light of Proposition 4.8, we assume without loss of generality that A t t 0 a t dt 1 {t≥τ A } , with a t : h t 1 {t≤τ} . Assume that min i v t, z i < 0 < max i v t, z i . Otherwise if, for instance, max i v t, z i ≤ 0, then it can be shown that h t ≥ 0, a.s. similarly to case b in Example 4.6 , and the first term of h is 0 under our convention that ∞ · 0 0. Notice that, in any case, one can find a predictable process z taking values on {z i } n i 1 , such that Write X · : · 0 G s dW s · 0 R 0 F s, z d N s, z for an F ∈ G loc N to be determined in the sequel. For ξ ≥ ξ it suffices to prove the existence of a field D satisfying both conditions below: International Journal of Stochastic Analysis then, F is defined as D F . Similarly, for ξ to belong to Γ it suffices that clearly nonnegative, b and c hold with equality. Moreover, the fact that inequalities c hold with equality implies that { ξ t V β t } 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 ϑ < 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 C resp., C such that ϑ z ≡ ϑ on C resp., ϑ z ≡ ϑ on C and 0 < ν C , ν C < ∞. Taking 4.42 b and c above will hold with equality.

Replicability of the Upper Bound
We now show that the tentative optimum final wealth V Γ z , suggested by the inequality iii in Theorem 3.5, is super-replicable. We will combine the dual optimality of ξ * with the superhedging theorem, which states that given a contingent claim H satisfying w : sup Q∈M E Q { H} < ∞, one can find for any fixed z ≥ w an admissible trading strategy β * depending on z such that V z,β * T ≥ H almost surely see Kramkov 2 , and also Delbaen and Schachermayer 1 . Recall that M denotes the class of all equivalent risk neutral probability measures. when I y z ξ * y z T < H.
Proof. For simplicity, we write ξ * t : ξ * y z t , y y z , and Fix an equivalent risk neutral probability measure Q ∈ M, and let ξ t dQ| F t /dP| F t be its corresponding density processes. Here, Q| F t resp., P| F t is the restriction of the measure Q resp., P to the filtration F t . 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 V β is a Plocal martingale necessarily nonnegative by admissibility , and thus, ξ is in Γ. 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 ∩ Γ and the fact that ξ * ∈ Γ see Theorem 4.1 , ξ ε : εξ 1 − ε ξ * belongs to Γ, for any 0 ≤ ε ≤ 1. Moreover, since U is convex and U y − I y ∧ H , The random variable yH ξ T ξ * T is integrable since by assumption w Γ < ∞. We can then apply dominated convergence theorem to get which is nonnegative by condition i in Theorem 3.5. Then, using condition ii in Theorem 3.5, Since Q ∈ M is arbitrary, sup Q∈M E Q V * ≤ z. By the super-hedging theorem, there is an admissible trading strategy β * for z such that The second statement of the theorem is straightforward since U z is strictly increasing on z < H.

18
International Journal of Stochastic Analysis

Concluding Remarks
We conclude the paper with the following remarks.

(i) The dual class Γ
The dual domain of the dual problem can be taken to be the more familiar class of equivalent risk-neutral probability measures M.
To be more precise, define Γ : Since Γ is obviously a convex subclass of Γ, Theorem 3.5 implies that, as far as for each z ∈ 0, w , there exist y : y z > 0 and ξ * : ξ * y z ∈ Γ 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 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 for all 0 ≤ ε ≤ 1, where ξ ε εξ 1 − ε 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 ξ n T → ξ * T a.s. Thus, ξ ε can be approximated by the elements ξ ε,n : εξ 1 − ε ξ n in Γ, for which we know that 6.5 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 * ∈ Γ of the dual problem induced by Γ Γ. We now further know that ξ * belongs to the class Γ ∩ Γ S defined in 4.3 , and hence, enjoys an explicit parametrization of the form 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: where W is a standard d-dimensional Brownian motion, N dt, dz is an independent Poisson random measure on R × R d , and N dt, dz N dt, dz − EN dt, dz . Consider a market model consisting of n 1 securities: one risk free bond with price dB t : r t B t dt, B 0 1, t ≥ 0, 6.8 and n risky assets with prices determined by the following stochastic differential equations with jumps: where the processes r, b, σ, and v are predictable satisfying usual integrability conditions cf. Kunita 10 . We assume that F : F ∞ − , where F : {F t } t≥0 is the natural filtration generated by W and N; namely, F t : σ W s , N 0, s × A : s ≤ t, A ∈ B R d . 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 wealthconsumption 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": International Journal of Stochastic Analysis where c is the instantaneous rate of consumption. To be more precise, the cumulative consumption at time t is given by C t : t 0 c u du and the discounted wealth at time t is given by 6.11 Here, U 1 is a state-dependent utility function and U 2 t, · is a utility function for each t. The dual problem can now be defined as follows: 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 ≤ ξ 0 ≤ 1 and {ξ t S t } t≤T is a supermartingale. The dual Theorem 3.5 can be extended for this problem. However, the existence of a wealthconsumption 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.

A. Convex Classes of Exponential Supermartingales
The goal of this part is to establish the theoretical foundations behind Theorem 4.1. We begin by recalling an important optional decomposition theorem due to Föllmer and Kramkov 12 . Given a family of supermartingales S satisfying suitable conditions, the result characterizes the nonnegative exponential local supermartingales ξ : ξ 0 E X − A , where X ∈ S and A ∈ V , in terms of the so-called upper variation process for S. Concretely, let P S be the class of probability measures Q ∼ P for which there is an increasing predictable process {A t } t≥0 depending on Q and S such that {X t − A t } t≥0 is a local supermartingale under Q, for all X ∈ S. The smallest of such processes A is denoted by A S Q and is called the upper variation process for S corresponding to Q. For easy reference, we state Föllmer and Kramkov's result see 12 for a proof .
Theorem A.1. Let S be a family of semimartingales that are null at zero, and that are locally bounded from below. Assume that 0 ∈ S, and that the following conditions hold: ii S is closed under theÉmery distance, iii P S / ∅, then, the following two statements are equivalent for a nonnegative process ξ: 1 ξ is of the form ξ ξ 0 E X − A , for some X ∈ S and an increasing process A ∈ V ; 2 ξ/E A S Q is a supermartingale under Q for each Q ∈ P S .

International Journal of Stochastic Analysis 21
The next result is a direct consequence of the previous representation. Recall that a sequence of processes {ξ n } n≥1 is said to be "Fatou convergent on π" to a process ξ if {ξ n } n≥1 is uniformly bounded from below and it holds that is convex and closed under Fatou convergence on any fixed dense countable set π of R ; that is, if {ξ n } n≥1 is a sequence in Γ S that is Fatou convergent on π to a process ξ, then ξ ∈ Γ S .
Proof. The convexity of Γ S is a direct consequence of Theorem A.1, since the convex combination of supermartingales remains a supermartingale. Let us prove the closure property. Fix a Q ∈ P S and denote C t : E A S Q . Notice that C t > 0 because A S Q t is increasing and hence, its jumps are nonnegative. Since ξ n ∈ Γ S , {C −1 t ξ n t } t≥0 is a supermartingale under Q. Then, for 0 < s < t , By Fatou's lemma and the right-continuity of process C, Finally, using the right continuity of the filtration, we have where 0 ≤ s < t. Since Q is arbitrary, the characterization of Theorem A.1 implies that ξ ∈ Γ S .
The most technical condition in Theorem A.1 is the closure property underÉmery distance. The following result is useful to deal with this condition. It shows that the class of integrals with respect to a Poisson random measure is closed with respect toÉmery distance, thus extending the analog property for integrals with respect to a fixed semimartingale due to Mémin 13 . 22 International Journal of Stochastic Analysis Theorem A.3. Let Θ be a closed convex subset of R 2 containing the origin. Let Π be the set of all predictable processes F, G , F ∈ G loc N , and G ∈ L 2 loc W , such that F t, · G t 0, for all t ≥ T , and F ω, t, z , G ω, t ∈ Θ, for P × dt × ν dz -a.e. ω, t, z ∈ Ω × R × R 0 . Then, the class S : is closed under convergence with respect toÉmery's topology.
Proof. Consider a sequence of semimartingales Let us extend M n and A n to 0, ∞ by setting M n t M n t∧T and A n A n t∧T for all t ≥ 0. Also, we extend Q for A ∈ F by setting Q A : A ξdP, so that Q ∼ P on F . In that case, it can be proved that A loc P A loc Q . This follows essentially from 17, Proposition III.3.5 and Doob's Theorem. Now, let ξ t : dQ| F t / dP| F t E ξ | F t , denote the density process. Since ξ is bounded, both {ξ t } t and {|Δξ t |} t are bounded. By 17, Lemma III.3.14 and Theorem III.3.11 , the P-quadratic covariation X n , ξ has P-locally integrable variation and the unique canonical decomposition M n A n of X n relative to Q is given by Also, the P-quadratic variation of the continuous part X n,c of X n relative to P , given by X n,c , X n,c · · 0 G n s 2 ds, is also a version of the Q-quadratic variation of the continuous part of X n relative to Q . By the representation theorem for local martingales relative to Z see, e.g., 17 A.14 as n, m → ∞. Using the notation Ω : Ω×R ×R and P : P×B R , where P is the predictable σ-field, we conclude that {F n } n≥1 is a Cauchy sequence in the Banach space International Journal of Stochastic Analysis such that G n → G, as n → ∞. In particular, F, G satisfies condition iv since Y 1 D is strictly positive, and each F n , G n satisfies iv . Also, F ∈ G loc N relative to Q in light of A loc P A loc Q . Similarly, · 0 G 2 s ds belongs to A loc Q , and hence, belongs to A loc P . It follows that the process X : is a well-defined local martingale relative to P. Applying Girsanov's Theorem to X relative to Q and following the same argument as above, the purely discontinuous local martingale and bounded variation parts of X are, respectively, A.18 The continuous part of X has quadratic variation · 0 G 2 s ds. We conclude that X ∈ M 2 Q ⊕ A Q and X n → X on M 2 Q ⊕ A Q . Then, X n converges underÉmery's topology to X and hence, X X.

B. Proofs of Some Standard Convex Duality Results
This appendix sketches the proofs of the results in Section 3. The proofs are standard in convex duality and are given only for the sake of completeness. Without loss of generality, one can assume that each process ξ n is constant on T, ∞ . By Lemma 5.2 in 12 , there exist ξ n ∈ conv ξ n , ξ n 1 , . . . , n ≥ 1, and a nonnegative supermartingale { ξ t } t≥0 with ξ 0 ≤ 1 such that {ξ n } n≥1 is Fatou convergent to ξ on the rational numbers for all t ≥ 0. By Fatou's Lemma, it is not hard to check that { ξ t V t } t≤T is a supermartingale for every admissible portfolio with value process V , and hence, ξ ∈ Γ. Next, since the ξ n 's are constant on T, ∞ and U ·; ω is convex, Fatou's Lemma implies that E U y ξ T , ω ≤ v Γ y Finally, we need to verify that, when Γ is convex, equality above is attained and that ξ can be approximated by elements of Γ. for all z < w Γ . Thus, f z · attains its minimum at some y z ∈ 0, ∞ . By Proposition 3.4, we can find a ξ y z ∈ Γ such that v y z E U y z ξ y z T , ω , B.8 proving the i above. Now, consider the function F u : uy z z E U uy z ξ y z T , u > 0. B.9 Since ξ y z can be approximated by elements in Γ, for each ε > 0 there exists a ξ yz v y − ε y z z E U y z ξ y z T − ε.

B.11
Since ε > 0 is arbitrary, the function F u attains its minimum at u 1. On the other hand,