This paper consists of two independent but related parts. In the first part (Section 3), we consider a class of backward stochastic differential equations (from now on BSDEs) (see (3.1) below) in the filtration generated by a Lévy process and, for the case of a Lipschitz generator of sublinear growth, we establish the existence and uniqueness of the solution in Theorem 3.3 below. We stress that the proof of Theorem 3.3 relies on the predictable representation property (from now on PRP) obtained by Di Tella and Engelbert in [6] (see also [5]) and recalled in Theorem 2.2 below.

A similar class of BSDEs has been considered by Nualart and Schoutens in [18]. Their approach is based on the PRP of the orthogonalized Teugels martingales, that was obtained in Nualart and Schoutens [17]. However, the PRP of the family of Teugels martingales requires an exponential moment of the Lévy measure outside the origin. On the other hand, for the PRP in [6] no additional assumptions on the Lévy process are needed. Hence, we are able to consider this class of BSDEs for general Lévy processes. Therefore, Theorem 3.3 below generalizes [18, Theorem 1] to arbitrary Lévy processes but with a great variety of systems of martingales in place of Teugels martingales.

The second part of the present paper (Section 4) is devoted to the study of a problem of logarithmic utility maximization of terminal wealth. We solve the problem having in mind the dynamical approach first introduced by Rouge and El Karoui in [19] and further developed by Hu, Imkeller and Müller in [10] in a Brownian setting, which is based on a martingale optimality principle constructed via BSDEs. In case of a more general filtration supporting martingales with jumps, the dynamical approach based on BSDEs has been followed by Morlais in [15, 16] and by Becherer in [1] to study the problem of exponential utility maximization of the terminal wealth.

The logarithmic utility maximization problem considered in this paper is analogous to the one studied in [10, Section 4] for a Brownian filtration: We extend the results of [10, Section 4] to Lévy processes with both Gaussian part and jumps.

Because of the special structure of the logarithmic utility, it turns out that, in the present paper, the martingale optimality principle can be constructed in a direct and independent way, without using BSDEs. However, as we shall explain (see Remark 4.7 below), this can be also alternatively done using a BSDE of the form of (3.1) below.

In [8] (see also references therein) Goll and Kallsen have studied the problem of the logarithmic utility maximization in a very general context (that, in particular, recovers Lévy processes), combining duality methods and semimartingale characteristics calculus. In [8], the authors assume the convexity of the set in which the admissible strategies for the optimization problem take values. However, since for the dynamical approach this further assumptions is not needed, the set of the constraints considered in the present paper will be closed and non-necessarily convex. Other references for the logarithmic utility maximization by the duality approach are Goll and Kallsen [7] and Kallsen [14]. We also recall the work by Civitanić and Karatzas [2] about utility optimization by the duality approach.

General setting of the paper. Let (Ω,F,P) be a complete probability space and F be a filtration satisfying the usual conditions. We fix a finite time horizon T>0 0$]]>. We only consider R -valued stochastic processes on the time interval [0,T] . When we say that a process X is a martingale we implicitly assume that X is càdlàg. We denote by P the σ-algebra of predictable subsets of [0,T]×Ω .

By H2 we denote the space of square integrable martingales X on [0,T] such that X0=0 . The space (H2,‖·‖2) endowed with the norm specified by ‖X‖22:=E[XT2] , for X∈H2 , is a Hilbert space. We observe that, by Doob’s inequality, the ‖·‖2 -norm is equivalent to the norm ‖X‖H22:=E[supt∈[0,T]Xt2] , X∈H2 . However, (H2,‖·‖H2) is not a Hilbert space.

For X∈H2 , we denote by ⟨X,X⟩ the predictable square variation process of X. We recall that ⟨X,X⟩ is the unique predictable increasing process starting at zero such that the process X2−⟨X,X⟩ is an F -martingale. If X,Y∈H2 , we define ⟨X,Y⟩ by polarization and say that X and Y are orthogonal if ⟨X,Y⟩=0 .

For X∈H2 , we introduce L2(X) as the space of F -predictable processes H such that ∫0THs2d⟨X,X⟩s is integrable. For H∈L2(X) , we denote by ∫0·HsdXs the stochastic integral of H with respect to X. It holds ∫0·HsdXs∈H2 , for every H in L2(X) . Furthermore, ∫0·HsdXs is characterized in the following way: A martingale Z∈H2 is indistinguishable from ∫0·HsdXs if and only if ⟨Z,Y⟩=∫0·Hsd⟨X,Y⟩s , for every Y∈H2 .

For two semimartingales X and Y, we denote by [X,Y] the covariation of X and Y, that is, we define [X,Y]t:=⟨Xc,Yc⟩+∑0≤s≤tΔXsΔYs , t∈[0,T] , where Xc and Yc denote the continuous local martingale part of X and Y, respectively.

A Lévy process with respect to F is an R -valued and F -adapted stochastically continuous càdlàg process L such that L0=0 , (Lt+s−Lt) is distributed as Ls and it is independent of Ft , for all s,t≥0 . In this case, we say that (L,F) is a Lévy process. By FL we denote the completion in F of the filtration generated by L. It is well known that FL satisfies the usual conditions (see [20]). Clearly, (L,FL) is again a Lévy process.

Let (L,F) be a Lévy process. We denote by μ the jump measure of L. Recall that μ is a homogeneous Poisson random measure on [0,T]×R with respect to F (see [13], Definition II.1.20). The F -predictable compensator of μ is deterministic and it is given by λ⊗ν , where λ is the Lebesgue measure on [0,T] and ν is the Lévy measure of L, that is, ν is, in particular, a σ-finite measure such that ν({0})=0 and x↦x2∧1 is ν-integrable. We call μ‾:=μ−λ⊗ν the compensated Poisson random measure associated with μ. By Bσ we denote the Gaussian part of L, which is an F -Brownian motion such that E[(Btσ)2]=σ2t , where σ2≥0 . By η∈R we denote the drift parameter of L and we call (η,σ2,ν) the characteristic triplet of L.

A function G on Ω˜:=[0,T]×Ω×R is said to be predictable if G is P⊗B(R) -measurable. Let G2(μ) denote the linear space of the predictable functions G on Ω˜ such that E[∑s≤TG2(s,ω,ΔLs(ω))1{ΔLs(ω)≠0}]<+∞ .

For G∈G2(μ) , we denote by ∫[0,·]×RG(s,y)μ‾(ds,dy) the stochastic integral of G with respect to μ‾ . Recall that ∫[0,·]×RG(s,y)μ‾(ds,dy) is defined as the unique purely discontinuous Z∈H2 with ΔZt(ω)=G(t,ω,ΔLt(ω))1{ΔLt(ω)≠0} (see [13], II.1.27).

Next, for any f∈L2(ν) , we introduce the martingale Xf∈H2 by Xtf:=∫[0,t]×Rf(y)μ‾(ds,dy),t∈[0,T].

Let (L,F) be a Lévy process with characteristic triplet (η,σ2,ν) on [0,T] . Then, for any f∈L2(ν) , the martingale Xf has the following properties:

(i) For every f∈L2(ν) , (Xf,F) is a Lévy process and E[(Xtf)2]=t∫Rf2(y)ν(dy)<+∞.

(ii) For every f,g∈L2(ν) , ⟨Xf,Xg⟩t=t∫Rf(y)g(y)ν(dy) .

(iii) For every f,g∈L2(ν) , Xf and Xg are orthogonal martingales if and only if f,g∈L2(ν) are orthogonal functions in L2(ν) .

Let T be an arbitrary subset of L2(ν) . Then we set (2.1) XT:={Bσ}∪{Xf,f∈T}.

As usual, ℓ2 denotes the Hilbert space of sequences a=(an)n≥1 of real numbers for which the norm ‖a‖ℓ22:=∑n=1∞(an)2 is finite.

Let T={fn,n≥1}⊆L2(ν) be an orthonormal basis. We remark that the orthogonal sum ∑n=1∞∫0·VsndXsfn converges in (and, therefore, belongs to) the Hilbert space (H2,‖·‖2) if and only if V=(Vn)n≥1 belongs to M2(ℓ2) . In this case we have the isometry ‖∑n=1∞∫0·VsndXsfn‖2=‖V‖M2(ℓ2) .

For the next theorem, we refer to [6, Section 4.2]. It states that the family XT , where T⊆L2(ν) is an orthonormal basis, possesses the PRP. Theorem 2.2.

Let (L,FL) be a Lévy process with characteristic triplet (η,σ2,ν) on the probability space (Ω,FTL,P) . Let T:={fn,n≥1} be an orthogonal basis of L2(ν) and let XT⊆H2 be the associated family of FL -martingales defined in (2.1). Then XT consists of pairwise orthogonal martingales and every FL -square integrable martingale N has the representation N=N0+∫0·ZsdBsσ+ ∑n=1∞∫0·VsndXsfn,Z∈L2(Bσ),Vn∈L2(Xfn),n≥1, where the spaces L2(Bσ) and L2(Xfn) are considered with respect to FL .

Let (L,FL) be a Lévy process with characteristic triplet (η,σ2,ν) . We consider the probability space (Ω,FTL,P) . Because of Theorem 2.2, for each orthogonal basis T={fn,n≥1} of L2(ν) , it is natural to consider the following BSDE: (3.1) Yt=ξ+∫tTf(s,Ys,Zs,Vs)ds−∫tTZsdBsσ− ∑n=1∞∫tTVsndXsfn, where f:[0,T]×Ω×R×R×ℓ2⟶R is a given random function, called the generator of BSDE (3.1), and ξ is an FTL -measurable random variable. We call the pair (f,ξ) the data of BSDE (3.1).

In [17] BSDE (3.1) is studied for Teugels martingales under the further assumption of the existence of an exponential moment for L, that is, E[eε|L1|]<+∞ , for an ε>0 0$]]>. We are going to study BSDE (3.1) for arbitrary Lévy processes L and for a great variety of families XT={Bσ}∪{Xfn,n≥1} , not restricting the analysis to Teugels martingales.

We introduce the space S2 by S2:={FL-adapted processesYsuch thatE[supt∈[0,T]Yt2]<+∞}.

We now state the assumptions on the generator f of (3.1) which we shall need in the proof of Theorem 3.3 below.

A generator f fulfilling the properties (i), (ii) and (iii) in Assumption 3.2, will be called admissible. Theorem 3.3.

Let ξ∈L2(Ω,FTL,P) and let f be an admissible generator. Then BSDE (3.1) with data (f,ξ) admits a unique solution (Y,Z,V) .