Forward Backward SDEs Systems for Utility Maximization in Jump Diffusion Models

We consider the classical problem of maximizing the expected utility of terminal net wealth with a final random liability in a simple jump-diffusion model. In the spirit of Horst et al. (2014) and Santacroce-Trivellato (2014), under suitable conditions the optimal strategy is expressed in implicit form in terms of a forward backward system of equations. Some explicit results are presented for the pure jump model and for exponential utilities.


Introduction
Portfolio's optimization and hedging problems are two classical problems which have been deeply investigated since the beginning of the seventies.Their mathematical formulation in continuous time was pioneered by Merton.In [17] he provides the strategy which maximizes the expected utility of a small investor in closed form by exploiting the Markovianity of the model for standard utilities.An alternative approach for not necessarily Markovian models was suggested by [2] and uses convex duality.This methodology has been developed in its full potential in [23], [7], [10], [3] and, in its modern form, in [11] and many others works thereafter.It is well known that convex duality and martingale methods lead to establish the existence and uniqueness of an optimal strategy in general market models and for non classical utility functions, but they do not provide a constructive characterization.A constructive form for the optimal portfolio has been determined for classical utilities in quite general market models by using dynamic programming techniques ( [9], [13]), [12]).Nevertheless, there exist just very few results for non classical utilities.A first work in this direction is given by [14] (see also [15]).In the non-Markovian framework, the problem is solved by dynamic programming and the optimal strategy is given in terms of the value function related to the problem and its derivatives.In turn, the value function is characterized as the solution of a backward stochastic partial differential equation under some assumptions imposed on the value function.Another subsequent work with a constructive characterization is [8], where the hedging problem is studied for general utilities in a Brownian model without resorting to dynamic programming.In this article the optimal strategy is described by means of the utility function with its derivatives and the solution of a fully coupled forward backward system.In [26] the same approach has been generalized to a continuous semimartingale setting.Further developments can be found in more recent works in Brownian settings (see [22] for a large investor problem with endogenous permanent market impacts and [27] for a stochastic maximum principle algorithm for constrained utilities maximization).In this paper, we consider a jump diffusion model driven by a Brownian motion and an independent simple Poisson process and study a classical problem of maximization for a general utility using the techniques introduced in [8] and [26].To our present knowledge, it is the first paper dealing with this approach in a model with jumps.We start by restating Proposition 2.1 in [26], which gives the derivative of the expectation of the terminal net wealth computed in an admissible strategy.The proof follows the same lines as in [26], but we add here an extra condition which was missing.The derivative is proved to be null for the optimal strategy and this represents our necessary condition for optimality.In our main result we use this key condition to express the optimal strategy in terms of a fully coupled forward-backward stochastic differential system.Besides, the vice versa of this result states that from a suitable solution of the system, the strategy which satisfies a certain optimality equation is admissible and optimal.As a result, in the jump diffusion model the optimal strategy is found to be implicitely defined by an equation written in terms of the solution of the forward backward system.Such optimality equation can not be explicitely solved even for the exponential utility, where the system decouples (see e.g., [18], [19], [20]).However, the optimal strategy admits an explicit expression in the pure jump model.In this setting, for the pure investment problem we also provide sufficient conditions in order to find a solution to the forward backward system, which is not in general an easy task.Moreover, the related optimal strategy turns out to have a more pleasant expression.Finally, we prove that these sufficient conditions are satisfied for exponential utilities, where an easy solution to the forward backward system and the corresponding optimal strategy are explicitly written.The paper is organized as follows.In Section 2, we introduce the model with the assumptions and state the preliminary proposition, containing the necessary condition for the optimality.In Section 3, we consider the jump diffusion model and give the characterization of the optimal strategy in terms of a fully-coupled forward backward system.In Section 4, we deal with the pure jump model.The forward backward characterization of the optimal strategy is followed by the application to the pure investment problem.Finally, the results of the previous sections are specified to exponential utilities in Section 5.

Model settings and preliminary results
We consider the classical problem of maximizing the expected utility of terminal net wealth, with a finite time horizon, when the asset price process is modeled by a jump-diffusion.Specifically, the financial model consists of a bank account which pays no interests and of one risky asset whose (discounted) price S is given by the stochastic exponential of a jump diffusion.On a probability space (Ω, F , P) we consider two independent processes W and N , both defined on [0, T ], where T < +∞ is the fixed time horizon: W is a standard (one-dimensional) Brownian motion and N a homogeneous Poisson process with intensity ν > 0. The probability space is equipped with the natural filtration generated by W and N (completed by the P-null sets of F ).For the sake of simplicity, we denote the filtration by F = (F t , t ∈ [0, T ]) and we suppose F T = F .We assume the price process S defined on the filtered probability space (Ω, F , F, P) with a dynamics given by where n t = N t − νt defines the compensated Poisson martingale.We suppose that the coefficients µ, σ, η are uniformly bounded predictable processes and, to ensure that S is almost surely positive, η > −1.Under these assumptions on the coefficients of the model there is no arbitrage in the market.We investigate the model in two main different cases: in the first one we consider σ 2 > 0, whereas the second concerns the pure jump model, i.e. σ 2 ≡ 0. We denote by π the dollar amount of risky asset in the portfolio and consider the wealth process X π which evolves according to the self-financing strategy π, i.e.
Moreover, let H be a bounded F T -measurable random variable, representing a liability due at time T .Given a utility function U defined on the real line, we consider the related problem of maximizing the expected utility of the terminal net wealth where x > 0 is a given initial capital and Π x is the set of admissible strategies defined as Taking into account the boundedness of µ, σ and η, this choice ensures the square integrability of the wealth process (2.2).Throughout the paper, the utility function U is assumed to be strictly increasing, strictly concave and three times continuously differentiable.Moreover, the following additional conditions are used as needed: < +∞, for a suitable random variable ξ; (H3) E[|U (ξ)|] < +∞, for a suitable random variable ξ.
From now on, according to (2.2) we consider the process X 0,h , where X 0,h 0 = 0 and h is a predictable bounded process, i.e.
Let us observe that, under our assumptions on the model and due to the boundedness of h, the pocess X 0,h is square integrable.
The following proposition represents the key starting point for the derivation of the main result and is a restatement of Proposition 2.1 in [26].For this reason, we omit the proof.We just remark that we add here an extra condition which was missing in [26].

Jump-diffusion model
In this section, we consider the dynamics of the price process (2.1) with σ 2 > 0.
The next two theorems are the main results of the paper.The first one characterizes the optimal strategy in terms of the solution of a forward-backward system of SDEs.The second theorem represents the vice versa and establishes the existence of an optimal strategy.
Theorem 3.1.Let π * ∈ Π x be optimal for the problem (2.3) and suppose (H1) holds.Under the assumptions of Proposition 2.1, there exists a smooth function G such that the following forward-backward system admits a solution (X, Y, Z, Ψ): where and the related optimal wealth process is equal to X.
Proof.The arguments used in this proof are similar to those in [8] and [26].We start by defining Since α is a (square integrable) martingale, it satisfies the backward stochastic differential equation (BSDE) where β and γ are respectively the predictable integrand appearing in the martingale representation of α, with respect to W and n.We consider the process Y , where with final value Y T = H.By Ito's formula we can write Observing that (3.9) can be immediately rewritten as After rearranging the dt, dW t and dn t terms together and replacing U ′−1 (α t ) = X π * t + Y t , we denote the integrands in the martingale terms respectively by Z t and Ψ t , i.e. we define the processes Therefore, we can see that Y solves the following BSDE If in the previous equation we replace β and Ψ using the expressions in (3.10), we obtain In order to obtain the BSDE (3.7) for Y we apply (2.5) of Proposition 2.1 to π * .
Using the integration by parts formula we have We now check that the first three integrals in (3.12) are martingales.
For the first integral we have where the constant C can differ from line to line.The first is Burkholder-Davis-Gundy inequality for p = 1, the third is Cauchy-Schwarz inequality, while the last is a consequence of Doob inequality and the boundedness of h, η and σ.Then the conclusion follows by the integrability assumptions on α.
Exploiting the same chain of inequalities, for the second integral we can write where the finiteness of the last expression is due to the square integrability of X 0,h and α.
We are left to prove that for the predictable process hηγ.This is true since h is bounded and by Cauchy-Schwarz Taking the expectation in (3.12) and recalling (2.5) we find where in the last equality we replaced β (3.14) gives the equation (3.15)Note that (3.15) includes the case η = 0 for which the optimal strategy can be explicitly written, as in [8].
For any ω ∈ Ω and t ∈ [0, T ], (3.15) can be seen as F = 0, where the function F : , and ∀π ∈ R, is continuously differentiable on its domain, because of the smoothness hypotheses on U .Moreover Since D is open, a global implicit function theorem can be applied (see Appendix).Consequently, for any w ∈ D the equation F (w, π) = 0 admits a unique solution π ∈ R and there exists a function G continuously differentiable on D and such that {(w, π) Thus, the optimal strategy is which is expressed in terms of the solution (Y, X, Z, Ψ) of the forward backward system (3.6-3.7),where the backward equation (3.7) is obtained replacing (3.8) in (3.11).Besides, we note that (3.6) represents the optimal wealth process.
For the next result we require σ −1 to be bounded in order to get the admissibility of the strategy π * .
Theorem 3.2.Suppose a solution of system (3.6-3.7)exists with Z ∈ H 2 and G a smooth function such that F (w, G(w)) = 0, where F is defined by (3.16).Assume (H1) and (H2), (H3) with ξ = X T + H.Moreover, let U ′ (X + Y ) be a positive martingale such that the stochastic integral ) is a square integrable martingale.Then the optimal strategy π * exists in the class Π x and is given by (3.8).
which, using (3.14), can be rewritten as (3.17) Since we have assumed that ) is a square integrable martingale, by the martingale representation theorem and the hypotheses on boundedness of η, µ and σ −1 we deduce that Using this fact, together with the hypotheses (H1) on ARA and Z ∈ H 2 , since (3.14) can be rewritten as we get that π * ∈ Π x .Now we are are left to show that π * is optimal.From (3.17), we observe that the positive martingale U ′ (X π * + Y ) can be written as the Doleans exponential of the martingale Since U is a concave function, for any π ∈ Π x we have that Thus we define the measure Q equivalent to P by and taking the expectation in (3.18) we write where the last equality is a consequence of the Girsanov Theorem.In fact, 1 Moreover, since π and π * are in Π x , we get that is a Q-local martingale.Finally, by the following chain of inequalities we see that it is a true Q-martingale: < +∞.
From (3.19), we have which means that π * is optimal.The proof is complete.

Remark 3.1. It is worth noting that assumption (H2) in Theorem 3.2 which requires
Therefore, it can be replaced by any other condition which ensures this requirement.

Pure jump model
In this section we study the case where the asset price is a pure jump process and show that for this model π * can be written in an explicit form in terms of the forward backward SDE system solution.We consider (2.1) with σ 2 ≡ 0, that is the price S is modeled by a pure jump process whose dynamics is dS t = S t − (µ t dt + η t dn t ), S 0 > 0.
We recall that the filtration F = (F t , t ∈ [0, T ]) is generated by a standard Brownian motion W and a simple Poisson process N with intensity ν > 0.Moreover, the coefficients µ and η are predictable and bounded with η > −1 and with the additional assumption c 1 ≤ µ η ≤ c 2 < ν.Theorem 4.1.Let π * ∈ Π x be optimal for the problem (2.3) and suppose (H1) holds.Under the assumptions of Proposition 2.1, there exists a solution (X, Y, Z, Ψ) of the following forwardbackward system Moreover, π * takes on the form and the optimal wealth process is equal to X.
Vice versa, suppose a solution of the system (4.20-4.21)exists and assume (H1) and ( H2), (H3) with ξ = X T + H.Moreover, let U ′ (X + Y ) be a positive martingale.If the strategy (4.22) is in the class Π x then it is optimal.
Proof.Using the same arguments as those for the jump-diffusion case, we can write where For the converse, let us observe that the positive martingale U ′ (X π * + Y ) can be written as the Doleans exponential of the martingale whose predictable quadratic covariation with 1 Thus, by Girsanov Theorem, 1 S − • S is a martingale with respect to the P-equivalent measure Q defined by The conclusion for the optimality of π * follows as in Theorem 3.2.
As previously said, it is not easy to find a solution of the coupled system.In the next proposition, we consider the pure investment problem, i.e.H = 0, and find a sufficient condition for obtaining an explicit solution to the system (4.20 -4.21) with Z = 0.At the end of the next section, we show an example where this condition is satisfied.
Proposition 4.1.Let H = 0 and suppose (H1) holds.Consider the system where a ∈ H 2 and A t = − T t a s ds, with A 0 deterministic.If U ′ (X)e A is a positive martingale, then (X, Y ) gives a solution to the forward backward system (4.20 -4.21) with Z = 0.Moreover, under (H3) with ξ = X T the strategy is in Π x and it is optimal.
Proof.Applying Ito's formula to U ′ (X t )e At , and taking into account the definition of A t , we get where From (4.28) we get Y T = 0 and, by Ito's formula, using (4.30) we see that which corresponds to Therefore, Y satisfies (4.21) with Z = 0.The admissibility of π * follows from (H1), the assumptions on a t and on the model coefficients.
In order to check the optimality of the strategy, by the converse part of Theorem 4.1, we are left to prove that U ′ (X)e A is a positive martingale and E (U ′ (X T )) 2 < +∞.But this is true since, by (4.31), U ′ (X)e A is the Doleans exponential of the martingale s ν ds is bounded.In fact, using Novikov condition, E(2M ) is a uniformly integrable martingale and, thus, [E T (M )] 2 = E T (2M )e M T has finite expectation.

Exponential utility
In this last section we tailor the results obtained previously to the case of the exponential utility, which has been extensively studied in the literature (e.g., [4], [5], [21], [25]).It can be easily seen that with this particular choice of utility function, the forward-backward system decouples and the problem reduces to the study of a backward stochastic differential equation.For results for continuous price models, see, e.g., [24], [9] in Brownian setting, and [13], [18] and [12] for more general continuous semimartingale models.For models allowing jumps, we quote among others [19], [20], [1], and [16] for the pure jump model.We consider the exponential utility U (x) = −e −δx with risk aversion parameter δ ∈ (0, +∞) and a bounded random liability H.In this case the evolution for does not depend on the wealth process X π * and is given by the following backward equation with final condition Y T = H.Moreover, equation (3.15) in the proof of Theorem 3.1 for the optimal strategy π * can be rewritten as from which we can deduce that also the optimal strategy is independent on the wealth.Let us point out that, employing a notation similar to [18], the driver of the BSDE can be rewritten in the form where, for ψ ∈ R, [ψ] δ = ν δ e δψ − 1 − δψ .Then, using the results in [18], we deduce that there exists a solution to the BSDE with Y bounded and Z, Ψ ∈ H 2 .
In the pure jump case, the backward evolution of Y reduces to and the optimal strategy π * can be written in an explicit form as stated in the next proposition.
Proposition 5.1.A solution of the backward equation (5.32) exists with Y bounded and Z, Ψ ∈ H 2 and the strategy π * is in Π x .In addition, if e −δ(X π * +Y ) is a positive martingale and either Z or Ψ is bounded, then (5.33) is optimal.
Proof.The existence of a solution to (5.32) with Y bounded and Z, Ψ ∈ H 2 is proved in Theorem 1 of Antonelli and Mancini [1].This theorem requires two main assumptions on the driver f of (5.32), which we can rewrite as A1) f is measurable and predictable and satisfies dP ⊗ dt -a.e.
for a strictly positive predictable process λ such that essup ω T 0 λ t dt < +∞; where the process In order to check these assumptions we rewrite the driver f in the form (5.36) Since π = 0 ∈ R, from (5.36) we deduce the validity of the left inequality in (5.34).To prove the right inequality, we consider the function Using the explicit form of the driver in (5.32), we can rewrite F (ψ) as whose maximum is attained at ψ * = 1 δ ln 1 − µ ην and holds We deduce that in (5.34) we can take λ , which is a bounded predictable and strictly positive process, thus the assumption A1) is verified.Using again the explicit form of the driver, (5.35) can be rewritten in the form and A2) is easily verified choosing ζ z,ψ,ψ ′ = − µ ην , D 1 = −1 and a suitable constant D 2 , which can be found since µ η is bounded.From the existence of the solution with Y bounded and Z, Ψ ∈ H 2 and from (5.33), thanks to the standing assumptions on the model, we deduce that π * ∈ Π x .Finally, from the vice versa of Theorem 4.1, the optimality of π * is guaranteed if we assume that either Ψ or Z is bounded.In fact, (H1) and (H3) trivially hold true, whereas (H2) reduces to check that E e −2δ(X π * T +H) is finite.First we suppose Ψ bounded, so is π * .Taking into account also H boundedness, we get Remark 5.1.We notice that, in the case Z bounded, the martingality assumption on e −δ(X π * +Y ) can be omitted since it is automatically satisfied.
Remark 5.2.In [16], the exponential utility maximization problem is studied within a BSDE framework and in a Lévy-driven pure jump asset model.Existence of optimal strategies are proved using BMO arguments and assumptions on the solutions of the BSDEs involved which imply the boundedness of the strategies.Although our market model is simpler, Proposition 5.1 represents an attempt to establish conditions for the existence of optimal strategies possibly not bounded.
We conclude the section with an example for the pure investment problem, where the hypotheses of Proposition 4.1 are automatically satified.
Proposition 5.2.If H = 0 and µ η is deterministic then the strategy is in Π x and it is optimal.
Proof.We choose in Proposition 4.1, which is a positive bounded process by the assumptions on the model.Then, the processes X and Y become On the other hand, we can observe that (5.37) represents the unique choice for a in Proposition 4.1 which makes U ′ (X)e A a martingale.In fact, U ′ (X)e A is a martingale if and only if Finally, the strategy is Remark 5.3.We notice that in the proposition above the hypothesis on µ η can be weakened by assuming deterministic T 0 a s ds, with the choice of a in (5.37).

Appendix
This result is a generalization of Lemma 1 in [6].Then there exists a function G ∈ C 1 (A; R) such that F (x, G(x)) = 0, ∀x ∈ A.
Proof.We start by proving that for any x ∈ A there exists y(x) ∈ R such that F (x, y(x)) = 0.
In order to do this, let us suppose that condition 1 holds.We can proceed in the same way if condition 2 is assumed.Let us fix x ∈ A. If F (x, 0) = 0, there is nothing to check.Otherwise, by Lagrange mean value theorem, we have where 0 < y 1 (x) < |F (x,0)| g(x) and − |F (x,0)| g(x) < y 2 (x) < 0.Then, by the intermediate value theorem, there exists − |F (x,0)| g(x) < y(x) < |F (x,0)| g(x) such that F (x, y(x)) = 0. Let us observe that, since F (x, •) is strictly monotone, y(x) is univocally determined.We are left with the task of proving that the function G defined by G(x) = y(x) is continuously differentiable on A. Let us fix x 0 ∈ A. Since F (x 0 , G(x 0 )) = 0, by the implicit function theorem we can find a neighborhood U x 0 of x 0 and a function ϕ x 0 ∈ C 1 (U x 0 ; R) such that F (x, ϕ x 0 (x)) = 0, for any x ∈ U x 0 .Considering that also F(x, G(x)) = 0 and F is strictly monotone, we deduce that G = ϕ x 0 , therefore continuously differentiable, on U x 0 .Due to the arbitrariness of x 0 we can then conclude that G ∈ C 1 (A; R) and get the thesis.

Theorem 6 . 1 .
Let  A be an open subset of R n .Let us consider a function F ∈ C 1 (A × R; R) and suppose there exists a function g : A → R such that one of the following conditions holds:1.∂ ∂y F (x, y) > g(x) > 0, ∀(x, y) ∈ A × R, 2. ∂ ∂y F (x, y) < g(x) < 0 ∀(x, y) ∈ A × R.