BSDEs with monotone generator driven by Brownian and Poisson noises in a general filtration

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BSDEs with monotone generator driven by Brownian and Poisson noises in a general filtration

T Kruse, A Popier

To cite this version:

T Kruse, A Popier. BSDEs with monotone generator driven by Brownian and Poisson noises in a general filtration. 2015. �hal-01094836v3�

BSDEs with monotone generator driven by Brownian and Poisson noises in a general filtration

T. Kruse ^∗, A. Popier ^† June 9, 2015

Abstract

We analyze multidimensional BSDEs in a filtration that supports a Brownian motion and a Poisson random measure. Under a monotonicity assumption on the driver, the paper extends several results from the literature. We establish existence and uniqueness of solutions inL^pprovided that the generator and the terminal condition satisfy appro- priate integrability conditions. The analysis is first carried out under a deterministic time horizon, and then generalized to random time horizons given by a stopping time with respect to the underlying filtration. Moreover, we provide a comparison principle in dimension one.

Introduction

The notion of nonlinear backward stochastic differential equations (BSDEs for short) was introduced by Pardoux and Peng [33]. A solution of this equation, associated with aterminal valueξ and a generator or driver f(t, ω, y, z), is a couple of stochastic processes(Y_t, Z_t)_t≤T such that

Y_t=ξ+ Z T

t

f(s, Y_s, Z_s)ds− Z T

t

Z_sdW_s, (1)

a.s. for allt≤T, where W is a Brownian motion and the processes (Yt, Zt)t≤T are adapted to the natural filtration ofW.

In their seminal work [33], Pardoux and Peng proved existence and uniqueness of a solution under suitable assumptions, mainly square integrability of ξ and of the process (f(t, ω,0,0))_t≤T, on the one hand, and, the Lipschitz property w.r.t. (y, z) of the generator f, on the other hand. Since this first result, BSDEs have proved to be a powerful tool for formulating and solving a lot of mathematical problems arising for example in finance (see e.g. [2, 13, 36]), stochastic control and differential games (see e.g. [15, 16]), or partial differential equations (see e.g. [31, 32]).

∗University of Duisburg-Essen, Thea-Leymann-Str. 9, 45127 Essen, Germany, e-mail:

thomas.kruse@uni-due.de

†Université du Maine, Laboratoire Manceau de Mathématiques, Avenue Olivier Messiaen, 72085 Le Mans, Cedex 9, France, e-mail: alexandre.popier@univ-lemans.fr

We would like to thank two anonymous referees for helpful comments and suggestions. Thomas Kruse acknowledges the financial support from the French Banking Federation through the Chaire "Markets in Transition".

Main results

The aim of this paper is to establish existence and uniqueness of solutions to BSDE in a general filtration that supports a Brownian motionW and an independent Poisson random measureπ. We consider the following multi-dimensional BSDE:

Yt=ξ+ Z T

t

f(s, Ys, Zs, ψs)ds− Z T

t

Z

U

ψs(u)eπ(du, ds)− Z T

t

ZsdWs− Z T

t

dMs. (2) The solution is given by the usual triple(Y, Z, ψ)and also an orthogonal local martingaleM which can not be reconstructed by the integrals w.r.t. to the Brownian and Poisson noise.

We assume that the generator f is monotonic (one-sided Lipschitz continuous) w.r.t. the y-variable and Lipschitz continuous w.r.t. to zand ψ. Under the condition that the data ξ and f(t,0,0,0) are inL^p,p > 1, we provide existence and uniqueness results in L^p spaces (the precise defintion will be given in Section 1).

Further contributions are a comparison result in dimension one and existence and unique- ness when the terminal time is a non necessarily bounded stopping time.

Related literature

There are already a lot of works which provide existence and uniqueness results under weaker assumptions than the ones of Pardoux and Peng [33] or El Karoui et al [12]. A huge part of the literature focuses on weakening the Lipschitz property of the coefficient f w.r.t. the y-variable. For example, Briand and Carmona [4] and Pardoux [31] consider the case of a monotonic generator w.r.t. y with different growth conditions. There have been relatively few papers which deal with the problem of existence and uniqueness of solutions in the case where the coefficients are not square integrable. El Karoui et al. [13] and Briand et al. [5]

have proved existence and uniqueness of a solution for the standard BSDE (1) in the case where the data belong only to L^p for some p≥1.

Another strand of research in the theory of BSDEs concerns the underlying filtration.

In [33] the filtration is generated by the Brownian motionW. Since the work of Tang and Li [39], a lot of papers (see e.g. [1, 3, 26, 29, 37] or the books of Situ [38] or recently of Delong [9]) treat the case where the filtration is generated by the Brownian motionW and a Poisson random measure π independent ofW. In most of these papers, the generator f is supposed to be Lipschitz in y, even if the monotonic case is mentioned (see [37]) and all coefficients are square integrable. Yao [41] studies the L^p case, p > 1, and gives existence and uniqueness result in the case where the generator is monotone but with at most linear growth w.r.t. y. Li and Wei [24] give existence und uniqueness results for a fully coupled forward backward SDE under some monotonicity condition andL^p coefficients,p≥2. Note that this monotonicity condition involves the coefficients of the forward diffusion and is not the same as the assumption imposed on the generator in this paper. An extension to BSDEs driven by a continuous local martingale X and an integer-valued random measure π has been studied by Xia [40]. Xia supposes that the filtration satisfies the representation property with respect to X and π and that the driver is Lipschitz continuous and square integrable.

For more general filtrations, the representation property of a local martingale is no more true (see Section III.4 in [17]) and an additional (orthogonal) martingale term has to be introduced in the definition of a solution. This approach was developed in the seminal work of El Karoui and Huang [11] and by Carbone et al. [6] for càdlàg martingales. The filtration Fis supposed to be complete, right continuous and quasi-left continuous. For a given square integrable martingaleX(hXidenotes the predictable projection of the quadratic variation), the BSDE (1) becomes

Y_t=ξ+ Z _T

t

f(s, Y_s, Z_s)dhXi_s− Z _T

t

Z_sdX_s−M_T +M_t. (3) The solution is now the triple(Y, Z, M)where M is a square integrable martingale orthogo- nal toX. Øksendal and Zhang [28] analyse BSDE of the form (3) wheref does not depend on z, and apply to insider finance (see also Ceci et al. [7]). Liang et al. [25] also obtain results for a particular class of BSDE (3) on an arbitrary filtered probability space. In these papers, existence and uniqueness of the solution of (3) is proved for a Lipschitz continuous function f and under square integrability condition (in [28] the monotone case is treated butf does not depend onz). The Hilbertian structure ofL²(Ω,F_T,P)is used in Cohen and Elliott [8] (see also [19]). If L²(Ω,F_T,P) is a separable Hilbert space, then an orthogonal basis of martingales can be introduced instead of X and there is no orthogonal additional term M in (3). Z becomes a sequence of predictable processes. The special case of a Lévy noise is treated before by Nualart and Schoutens [27]: the orthogonal basis of martingales is explicitely given by the Teugels martingales.

In very recent papers, Klimsiak has developed the results concerning BSDEs in this general framework in two directions. First for reflected BSDE ([18, 19]), and secondly for parabolic equations ([20, 21]) with measure data.

Main contributions

Let us outline the main contributions of our paper compared to the existing literature.

First of all our paper generalizes many results from the works [1, 3, 26, 29, 37, 39] dealing with a filtration generated by the Brownian motion and the Poisson random measure since we allow for a more general filtration.

Moreover we provide existence and uniqueness of solutions in L^p-spaces, p > 1. In the case where the generator depends on the stochastic integrand w.r.t. a Poisson random measure, the case whenp <2 has to be handled carefully and can not be treated as in [5].

Indeed in this case Burkholder-Davis-Gundy inequality with p/2 < 1 does not apply and the L^p/2-norm of the predictable projection cannot be controlled by the L^p/2-norm of the quadratic variation (see Inequality (25) and [23]). Yao [41] obtains similar results but for a generator with at most linear growth w.r.t. y (and for a filtration generated byW and π).

Klimsiak [19] considers L^p solutions of BSDE, with p6= 2, in a general filtration but where the driver only depends ony.

Compared to [6] or [40], our assumptions are in some sense more restrictive as we assume that the continuous part of the given martingaleX of BSDE (3) is a Brownian motion W and the random measure associated to the jumps of X is a Poisson random measure π.

However we weaken the assumptions on the driver and on the terminal condition: the generator is only supposed to be monotone and the terminal condition is allowed to be only L^p-integrable. To the best of our knowledge, there is no existence and uniquenesss result for multi-dimensional BSDE withL^p coefficients in a general filtration. The generalization of our results for BSDE of the form (3) requires some sophisticated integrability conditions to take account of the predictable projectionhXiof the quadratic variation ofX. Therefore it is left for future research.

Moreover we provide a comparison principle and existence and uniqueness in the case with random terminal time for BSDE of type (2). The proof of the comparison principle generalizes the arguments of [35] to the situation where the filtration is not only generated by Brownian and Poisson noise.

Finally our setting is important for the control problem we study in the paper [22] (see also [14]). The control problem arises in mathematical finance and models the optimal liquidation of a financial position in an illiquid market. In [14] the authors consider the case when the filtration is generated by a Brownian motion and a independent Poisson measure.

In [22] we do not impose any condition on the filtration generated by the market (except right-continuity, completeness and quasi-left continuity) and the Poisson random measure represents the limit orders of the trading strategy. The optimal strategy is determined by a solution of a BSDE of the form (2) where the generator depends on the Poisson random measure.

Decomposition of the paper

The paper is decomposed as follows. In the first section, we give the mathematical setting and the main results of this paper. In the second part, we consider square integrable coefficients and we prove existence and uniqueness of the solution. To prove it we mainly follow the scheme of [31] with suitable modifications. In the next part, we extend the result to L^p coeffcients for any p > 1. For p > 2, the existence is derived from the existence in the L² case with the right a priori estimate. For 1< p < 2, an extra computation has to be made since the functionx7→ |x|^p is not smooth in this case. We have to extend Lemma 2.2 in [5] or Proposition 2.1 in [18] to our framework. In the last two sections, we add two extensions: the comparison result in dimension one, and existence and uniqueness when the terminal time is a stopping time. Note that the comparison principle requires an extra condition when the generator depends on the jump part (see the counterexample in [1]).

But instead of using Girsanov’s theorem to obtain the comparison between two solutions, we generalize the argument of [35], which is less restrictive. This point will be crucial in [22].

1 Settings and main results

Let us consider a filtered probability space(Ω,F,P,F= (F_t)_t≥0). The filtration is assumed to be complete, right continuous and quasi-left continuous, which means that for every sequence (τ_n) of F stopping times such that τ_n ր τ˜ for some stopping time τ˜ we have W

n∈NF_τn =F_τ˜.. Without loss of generality we suppose that all semimartingales have right

continuous paths with left limits. We assume that (Ω,F,P,F = (F_t)_t≥0) supports a k- dimensional Brownian motion W and a Poisson random measure π with intensity µ(du)dt on the spaceU ⊂R^m\ {0}. The measure µis σ-finite onU such that

Z

U

(1∧ |u|²)µ(du) <+∞.

The compensated Poisson random measure eπ(du, dt) =π(du, dt)−µ(du)dt is a martingale w.r.t. the filtrationF.

In this paper for a given T ≥0, we denote:

• P: the predictable σ-field on Ω×[0, T]and Pe=P ⊗ B(U) where B(U) is the Borelianσ-field on U.

• OnΩ = Ωe ×[0, T]× U, a function that isPe-measurable, is called predictable. G_loc(π) is the set ofPe-mesurables functions ψ onΩe such that for any t≥0a.s.

Z t

0

Z

U

(|ψ_s(u)|²∧ |ψ_s(u)|)µ(du)<+∞.

• D(resp. D(0, T)): the set of all predictable processes onR₊(resp. on[0, T]). L²_loc(W) is the subspace of Dsuch that for any t≥0 a.s.

Z _t

0

|Z_s|²ds <+∞.

• M_loc: the set of càdlàg local martingales orthogonal to W andπ. Ife M ∈ M_locthen [M, Wⁱ]_t= 0,1≤i≤k [M,eπ(A, .)]_t= 0

for all A ∈ B(U). In other words, E(∆M ∗π|Pe) = 0, where the product ∗ denotes the integral process (see II.1.5 in [17]). Roughly speaking, the jumps ofM andπ are independent.

• M is the subspace of M_loc of martingales.

We refer to [17] (see also [3]) for details on random measures and stochastic integrals. As explained above, the filtrationFsupports the Brownian motionW and the Poisson random measureπ.

Lemma 1 (Lemma III.4.24 in [17]) Every local martingale has a decomposition Z _.

0

Z_sdW_s+ Z _.

0

Z

U

ψ_s(u)eπ(du, ds) +M where M ∈ M_loc, Z ∈L²_loc(W), ψ∈G_loc(µ).

Now to define the solution of our BSDE, let us introduce the following spaces forp≥1.

• D^p(0, T) is the space of all adapted càdlàg processes X such that E sup

t∈[0,T]

|X_t|^p

!

<+∞.

For simplicity,X_∗ = sup_t∈[0,T]|X_t|.

• H^p(0, T) is the subspace of all processes X∈ D(0, T) such that E

"Z T 0

|X_t|²dt ^p/2#

<+∞.

• M^p(0, T) is the subspace of Mof all martingales such that Eh

([M]_T)^p/2i

<+∞.

• L^p_π(0, T) =L^p_π(Ω×(0, T)× U): the set of processesψ∈G_loc(µ)such that E

"Z T 0

Z

U

|ψ_s(u)|²µ(du)ds p/2#

<+∞.

• L^p_µ=L^p(U, µ;R^d): the set of measurable functions ψ:U →R^d such that kψk^p_Lp

µ = Z

U

|ψ(u)|^pµ(du)<+∞.

• T : the set of all finite stopping times andT_T the set of all stopping times with values in[0, T].

• E^p(0, T) =D^p(0, T)×H^p(0, T)×L^p_π(0, T)×M^p(0, T).

We consider the following BSDE (2):

Y_t=ξ+ Z T

t

f(s, Y_s, Z_s, ψ_s)ds− Z T

t

Z

U

ψ_s(u)eπ(du, ds)− Z T

t

Z_sdW_s− Z T

t

dM_s. Here, the random variable ξ isF_T-measurable with values inR^d(d≥1) and the generator f : Ω×[0, T]×R^d×R^d×k×L²

µ → R^d is a random function, measurable with respect to P rog× B(R^d)× B(R^d×k)× B(L²_µ)whereP rog denotes the sigma-field of progressive subsets ofΩ×[0, T].

The unknowns are (Y, Z, ψ, M) such that

• Y is progressively measurable and càdlàg with values in R^d;

• Z ∈L²_loc(W), with values in R^d×k;

• ψ∈G_loc(µ) with values in R^d;

• M ∈ M_loc with values inR^d.

OnR^d,|.|denotes the Euclidean norm andR^d×kis identified with the space of real matrices with drows and k columns. If z ∈R^d×k, we have |z|² =Trace(zz^∗). IfM is a R^d-valued martingale inM, the bracket process[M]_tis

[M]_t= Xd

i=1

[Mⁱ]_t, where Mⁱ is the i-th component of the vector M.

Throughout the paper, the following assumptions on the generator f are denoted by (Hex).

(H1) For everyt∈[0, T],z∈R^d×k and everyψ∈L²_µthe mapping y∈R^d7→f(t, y, z, ψ)is continuous. Moreover there exists a constant α such that

hf(t, y, z, ψ)−f(t, y^′, z, ψ), y−y^′i ≤α|y−y^′|².

(H2) For every r > 0 the mapping (ω, t) 7→ sup_|y|≤r|f(t, y,0,0) −f(t,0,0,0)| belongs to L¹(Ω×[0, T],P⊗m).

(H3) f is Lipschitz continuous w.r.t. z and ψ: there exists a constantK such that for any tand y, for any z, z^′ ∈Rand ψ, ψ^′ inL²(µ)

|f(t, y, z, ψ)−f(t, y, z^′, ψ^′)| ≤K(|z−z^′|+kψ−ψ^′kL²

µ).

We can suppose w.l.o.g. that α = 0. Indeed if (Y, Z, ψ, M) is a solution of (2) then ( ¯Y ,Z,¯ ψ,¯ M)¯ with

Y¯_t=e^αtY_t, Z¯_t=e^αtZ_t, ψ¯_t=e^αtψ_t, dM¯_t=e^αtdM_t satisfies an analogous BSDE with terminal condition ξ¯=e^αTξ and generator

f¯(t, y, z, ψ) =e^αtf(t, e^−αty, e^−αtz, e^−αtψ)−αy.

f¯satisfies assumptions (Hex) withα = 0. Hence in the rest of this paper, we will suppose thatα= 0.

Our main results can be summarized as follows. Under Assumptions (Hex) and if for some p >1

E

|ξ|^p+ Z T

0

|f(t,0,0,0)|^pdt

<+∞,

there exists a unique solution (Y, Z, ψ, M) in E^p(0, T) to the BSDE (2). The comparison principle holds for this BSDE. Moreover with a suitable conditions (see (H5’) and (H6)) the terminal timeT can be replaced by a stopping time τ.

2 L

solutions

Let us begin with the following definition:

Definition 1 (L²-solution) We say that (Y, Z, ψ, M) is a L²-solution of the BSDE (ξ, f) on [0, T]if

• (Y, Z, ψ, M) ∈ E²(0, T);

• and Equation (2)is satisfied P⊗dt-a.s.

In the rest of this section, a solution (Y, Z, ψ, M) will be supposed to be in E²(0, T). Now we want to prove existence of the solution of the BSDE with data(ξ, f). For this purpose, we will add the integrability conditions:

(H4) E(|ξ|²) +E Z T

0

|f(t,0,0,0)|²dt <+∞.

Some a priori estimates are needed. Note that the following results are modifications of the results obtained in [29].

Lemma 2 Let (Y, Z, ψ, M) ∈ E²(0, T) be a solution of BSDE (2). Then E

Z T 0

|Z_s|²ds+ Z T

0

Z

U

|ψ_s(u)|²µ(du)ds+ [M]_T

≤ CE sup

t∈[0,T]

|Y_t|²+ Z T

0

|f(t,0,0,0)|²dt

!

for some constant C depending only on K² andT. Proof. Letτ ∈ TT and by Itô’s formula on|Yt|²:

|Y_τ∧t|²+ Z τ

τ∧t

|Z_s|²ds+ Z τ

τ∧t

Z

U

|ψ_s(u)|²µ(du)ds+ [M]_τ−[M]_τ∧t

=|Y_τ|²+ 2 Z τ

τ∧t

Y_sf(s, Y_s, Z_s, ψ_s)ds−2 Z τ

τ∧t

Y_s⁻Z_sdW_s

−2 Z τ

τ∧t

Y_s⁻dM_s− Z τ

τ∧t

Z

U

|Y_s⁻+ψ_s(u)|²− |Y_s⁻|² e

π(du, ds). (4) But from (H1) and (H3):

yf(t, y, z, ψ) ≤ |y|

K|z|+KkψkL²

µ+|f(t,0,0,0)|

. Hence witht= 0 and Young’s inequality:

1 2

Z τ

0

|Zs|²ds+ 1 2

Z τ

0

Z

U

|ψs(u)|²µ(du)ds+ [M]τ

≤((4K²+ 1)T + 1) sup

t∈[0,T]

|Y_t|²+ Z T

0

|f(s,0,0,0)|²ds−2 Z τ

τ∧t

Y_s⁻Z_sdW_s

−2 Z τ

0

Y_s⁻dM_s− Z τ

0

Z

U

|Y_s⁻+ψ_s(u)|²− |Y_s⁻|² e

π(du, ds).

Moreover from the assumptions onY,Z,ψ andM, the stochastic integral terms w.r.t.W, M andπeare martingales. Now take τ =T and we can take the expectation on both parts:

E 1

2 Z T

0

|Z_s|²ds+1 2

Z T

0

Z

U

|ψ_s(u)|²µ(du)ds+ [M]_T

≤(4K²+ 2)TE sup

t∈[0,T]

|Y_t|²

! +E

Z T

0

|f(s,0,0,0)|²ds

which achieves the proof.

Lemma 3 Let (Y, Z, ψ, M) be a solution of BSDE (ξ, f) with the same conditions as in Lemma 2. Then

E sup

t∈[0,T]

|Y_t|²

!

≤ CE

|ξ|²+ Z T

0

|f(t,0,0,0)|²dt

for some constant C depending only on K andT.

Proof. For0≤t≤T, let us apply Itô formula (4) toe^βs|Y_s|² whereβ will be chosen later.

We have:

e^βt|Y_t|² = e^βT|Y_T|²+ 2 Z _T

t

e^βsY_sf(s, Y_s, Z_s, ψ_s)ds

− Z T

t

βe^βs|Y_s|²ds− Z T

t

Z

U

e^βs|ψ_s(u)|²µ(du)ds

− Z _T

t

e^βsd[M]_s− Z _T

t

e^βs|Z_s|²ds

− 2 Z T

t

e^βsY_s⁻dMs−2 Z T

t

e^βsY_s⁻ZsdWs

− Z _T

t

e^βs Z

U

|Y_s⁻+ψ_s(u)|²− |Y_s⁻|² e

π(du, ds).

From the assumptions on f, we have for any ε >0

Y_sf(s, Y_s, Z_s, ψ_s) = Y_s(f(s, Y_s, Z_s, ψ_s)−f(s,0, Z_s, ψ_s)) +Y_s(f(s,0, Z_s, ψ_s)−f(s,0,0, ψ_s)) + Y_s(f(s,0,0, ψ_s)−f(s,0,0,0)) +Y_sf(s,0,0,0)

≤ |Ys||f(s,0,0,0)|+K|Ys|kψskL²

µ+K|Ys||Zs|

≤ (1 + 2K²)

2ε |Y_s|²+ε

2(|f(s,0,0,0)|²+kψ_sk²L²

µ+|Z_s|²). (5)

We takeε= 1/2 and we obtain:

e^βt|Y_t|²+1 2

Z _T

t

e^βs|Z_s|²ds+1 2

Z _T

t

e^βskψ_sk²L²

µds+1 2

Z _T

t

e^βsd[M]_s

≤e^βT|Y_T|²+1 2

Z T

t

e^βs|f(s,0,0,0)|²ds

− Z _T

t

β−2(1 + 2K²)

e^βs|Y_s|²ds+ Γ_t,T

where Γ_t,.is a local martingale starting at zero at time t. Fixβ = 2(1 + 2K²)and we have:

e^βt|Y_t|²+1 2

Z T

t

e^βskψ_sk²L²

µds+1 2

Z T

t

e^βsd[M]_s

≤e^βT|Y_T|²+1 2

Z T

t

e^βs|f(s,0,0,0)|²ds+ Γ_t,T. (6) Since all local martingales are true martingales, we deduce that

sup

t∈[0,T]

E|Y_t|²+E Z T

0

|Z_s|²ds+E Z T

0

kψ_sk²L²

µds+E[M]_T

≤CE

|ξ|²+ Z _T

0

|f(t,0,0,0)|²dt

. (7)

Now withβ = 2(1 + 2K²) we write the Itô formula in a different way:

e^βt|Y_t|² = e^βT|Y_T|²+ 2 Z T

t

e^βsY_sf(s, Y_s, Z_s, ψ_s)ds

− Z _T

t

βe^βs|Y_s|²ds− Z _T

t

Z

U

e^βs|ψ_s(u)|²π(du, ds)

− Z T

t

e^βsd[M]_s− Z T

t

e^βs|Z_s|²ds

− 2 Z _T

t

e^βsY_s⁻dM_s−2 Z _T

t

e^βsY_s⁻Z_sdW_s

− 2 Z T

t

e^βs Z

U

Y_s⁻ψ_s(u)eπ(du, ds) which gives with (5):

e^βt|Y_t|² ≤e^βT|Y_T|²+ 1 2

Z T

t

e^βs|f(s,0,0,0)|²ds+1 4

Z T

t

e^βskψ_sk²L²

µds

−2 Z T

t

e^βsY_s⁻dM_s−2 Z T

t

e^βsY_s⁻Z_sdW_s−2 Z T

t

e^βs Z

U

Y_s⁻ψ_s(u)eπ(du, ds) (8) Next we apply the Burkholder-Davis-Gundy inequality to obtain

E sup

t∈[0,T]

Z _t

0

e^βsY_s⁻dM_s ≤cE

Z _T

0

e^2βs|Y_s⁻|²d[M]_s 1/2

≤ 1

8E sup

t∈[0,T]

e^βt|Y_t|²

!

+ 2c²E Z _T

0

e^βsd[M]_s. (9) By the same arguments we have

E sup

t∈[0,T]

Z _t

0

e^βsY_s⁻Z_sdW_s ≤cE

Z T 0

e^2βs|Y_s⁻|²|Z_s|²ds 1/2

≤ 1

8E sup

t∈[0,T]

e^βt|Y_t|²

!

+ 2c²E Z T

0

e^βs|Z_s|²ds. (10)

Finally the same result holds for the martingale Z .

0

Z

U

(Y_s−ψ_s(u))π(du, ds),e with

E sup

t∈[0,T]

Z t

0

e^βs Z

U

(Y_s−ψ_s(u))eπ(du, ds) ≤cE

Z T 0

e^2βs|Y_s−|² Z

U

|ψ_s(u)|²π(du, ds) 1/2

≤ 1

8E sup

t∈[0,T]

e^βt|Yt|²

!

+ 2c²E Z T

0

e^βskψsk²L²

µds. (11)

Now coming back to (8), and using estimates (9), (10), (11) and (7), we deduce that there existsC depending on K and T such that

E sup

t∈[0,T]

|Y_t|²

!

≤ CE

|ξ|²+ Z T

0

|f(t,0,0,0)|²dt

.

This achieves the proof.

The next result is an extension of the Proposition 2.1 in [4]. For convenience let us give the result and the proof.

Lemma 4 Let (Y, Z, ψ, M) ∈ E²(0, T) be a solution of BSDE (2) with bounded terminal condition ξ and generator|f(t,0,0,0)|: there exists a constantκ such that a.s.

sup

t∈[0,T]

|f(t,0,0,0)|+|ξ| ≤κ. (12)

Then Y is also almost surely bounded: there exists a constant β = 2(1 + 2K²) such that almost surely and for any t∈[0, T]

|Y_t|² ≤κ²e^β(T−t)

1 + 1 2β

.

Proof. We use Inequality (6) and since the involved local martingale Γ is a martingale, taking the conditional expectation w.r.t.F_t leads to: a.s. for everyt∈[0, T]

|Yt|²≤E

e^β(T−t)|ξ|²+1 2

Z T

t

e^β(s−t)|f(s,0,0,0)|²ds Ft

.

HenceY ∈ D^∞(0, T).

Now we prove a stability result.

Lemma 5 Let now(ξ, f) and(ξ^′, f^′) be two sets of data each satisfying the above assump- tions(Hex) and (H4). Let (Y, Z, ψ, M) (resp. (Y^′, Z^′, ψ^′, M^′)) denote a L²-solution of the BSDE (2) with data(ξ, f) (resp. (ξ^′, f^′)). Define

(Y ,b Z,b ψ,b M ,c ξ,bfb) = (Y −Y^′, Z−Z^′, ψ−ψ^′, M−M^′, ξ−ξ^′, f−f^′).

Then there exists a constant C depending onK² and T, such that E sup

t∈[0,T]

|Yb_t|²+ Z _T

0

|Zb_s|²ds+ Z _T

0

Z

U

|ψb_s(u)|²µ(du)ds+ [M]c_T

!

≤CE

|ξ|b²+ Z _T

0

|fb(t, Y_t^′, Z_t^′, ψ^′_t)|²dt

.

As a consequence of this lemma, we obtain uniqueness of the solution (Y, Z, ψ, M) for the BSDE (2) in the setE²(0, T) (see also Corollary 2 in dimensiond= 1).

Proof. Letτ ∈ T_T and by Itô’s formula on|Yb_t|²:

|Yb_τ∧t|²+ Z _τ

τ∧t

|Zb_s|²ds+ Z _τ

τ∧t

Z

U

|ψb_s(u)|²µ(du)ds+ [M]c_τ−[Mc]_τ∧t

=|bY_τ|²+ 2 Z τ

τ∧t

Yb_s(f(s, Y_s, Z_s, ψ_s)−f^′(s, Y_s^′, Z_s^′, ψ^′_s))ds−2 Z τ

τ∧t

Yb_s⁻Zb_sdW_s

−2 Z _τ

τ∧t

Yb_s⁻dMc_s− Z _τ

τ∧t

Z

U

|Yb_s⁻+ψb_s(u)|²− |Yb_s⁻|² e

π(du, ds).

From the monotonicity assumption on the generator and Young’s inequality, we have:

|Yb_τ∧t|²+ 1 2

Z τ

τ∧t

|Zb_s|²ds+1 2

Z τ

τ∧t

Z

U

|ψb_s(u)|²µ(du)ds+ [M]c_τ−[cM]_τ∧t

≤ |Yb_τ|²+ (4K²+ 1) Z _τ

τ∧t

|bY_s|²ds+ Z _τ

τ∧t

|fb(s, Y_s^′, Z_s^′, ψ^′_s)|²ds−2 Z _τ

τ∧t

Yb_s⁻Zb_sdW_s

−2 Z τ

τ∧t

Yb_s⁻dMcs− Z τ

τ∧t

Z

U

|Yb_s⁻+ψbs(u)|²− |Yb_s⁻|² e

π(du, ds). (13) Withτ =T and Gronwall’s lemma, we have for anyt∈[0, T]

E|bY_t|²≤CE

|ξ|b²+ Z _T

0

|fb(s, Y_s^′, Z_s^′, ψ_s^′)|²ds

.

Then using (13) with t= 0 and τ =T and the previous inequality we obtain E

Z T 0

|Zb_s|²ds+ Z T

0

Z

U

|ψb_s(u)|²µ(du)ds+ [Mc]_T

≤CE

|bξ|²+ Z T

0

|fb(s, Y_s^′, Z_s^′, ψ^′_s)|²ds

. Finally take the conditional expectation w.r.t. F_t in (13), the supremum over t ∈ [0, T] on both sides and applying Doob’s inequality to the supremum of the (F_τ∧t, t ∈ [0, T]) martingale on the right-hand side, we have:

E sup

t∈[0,T]

|Yb_t|²

!

≤CE

|bξ|²+ Z T

0

|fb(s, Y_s^′, Z_s^′, ψ_s^′)|²ds

.

This completes the proof.

Let us modify a little the growth assumption (H2):

(H2’) For every(t, y)∈[0, T]×R^d,|f(t, y,0,0)| ≤ |f(t,0,0,0)|+ϑ(|y|)where ϑ:R₊→R₊ is a deterministic continuous increasing function.

Now we can prove the following result.

Proposition 1 Under assumptions (H1)-(H2’)-(H3) and (H4), there exists a unique L²- solution(Y, Z, ψ, M) for the BSDE (2).

Proof. The proof follows closely the arguments in [20] and [29] (see also [6] or [40] for the Lipschitz case). Therefore we only sketch it.

• Step 1: we assume thatf is Lipschitz with w.r.t.y: there exists a constantK^′ such that for all(t, y, y^′, ψ)

|f(t, y, z, ψ)−f(t, y^′, z, ψ)| ≤K^′|y−y^′|. (14) Moreoverξ and f(t,0,0,0) satisfy the condition (12).

Under these assumptions, for(U, V, φ, N)inE²(0, T), we define the following processes (Y, Z, ψ, M) as follows:

Y_t=E

ξ+ Z T

0

f(s, U_s, V_s, φ_s)ds F_t

− Z t

0

f(s, U_s, V_s, φ_s)ds, and the local martingale

E

ξ+ Z T

0

f(s, U_s, V_s, φ_s)ds F_t

−Y₀ can be decomposed in three parts (see Lemma 1):

E

ξ+ Z T

0

f(s, Us, Vs, φs)ds Ft

−Y0 = Z t

0

ZsdWs+ Z t

0

ψs(u)π(du, ds) +e Mt

where Z ∈L²_loc(W) a.s., ψ ∈ G_loc(π) and M ∈ M_loc. From the conditions imposed on f and ξ, it is straightforward to prove that (Y, Z, ψ, M) ∈ E²(0, T). Moreover (Y, Z, ψ, M) is the unique solution of the BSDE

Y_t=ξ+ Z _T

t

f(s, U_s, V_s, φ_s)ds− Z _T

t

Z_sdW_s− Z _T

t

Z

U

ψ_s(u)πe(du, ds)− Z _T

t

dM_s. Therefore we may define the mapping Ξ :E²(0, T)→ E²(0, T)by putting

Ξ((U, V, φ, N)) = (Y, Z, ψ, M).

By standard arguments (see e.g. the proof of Theorem 55.1 in [29]) we can prove that Ξis contractive on the Banach space(E²(0, T),k.k_β)where

k(Y, Z, ψ, M)k_β = E (

sup

0≤t≤T

e^βt|Y_t|²+ Z T

0

e^βt|Z_s|²ds +

Z T

0

e^βt Z

U

|ψ_t(u)|²µ(du)dt+ Z .

0

e^βtdM_t

T

, with suitable constant β > 0. Consequently, Ξ has a fixed point (Y, Z, ψ, M) ∈ E²(0, T). Therefore,(Y, Z, ψ, M) is the unique solution of the BSDE (2).

• Step 2: We now show how to dispense with the assumptions (14) and (12). The main result is the following.

Lemma 6 Under assumptions (Hex) and (H4), given (V, φ) ∈ H²(0, T)×L²_π(0, T) there exists a unique process (Y, Z, ψ, M) in E²(0, T) such that

Y_t=ξ+ Z T

t

f(s, Y_s, V_s, φ_s)ds− Z T

t

Z_sdW_s− Z T

t

Z

U

ψ_s(u)eπ(du, ds)− Z T

t

dM_s. (15) The processf(s, y, V_s, φ_s) will be denoted byf(s, y).

First we keep the boundness condition (12) and we construct of smooth approximations (f_n, n∈N) of f (see proof of Proposition 2.4 in [30]). For any n,f_n is smooth and monotone in y, and thus locally Lipschitz in y. We cannot directly apply the Step 1 since f_n is not necessarily globally Lipschitz. But we just add a truncation function q_p inf_n:

fn,p(t, y) =fn(t, qp(y)), qp(y) =py/(|y| ∨p).

From the first step there exists a solution (Y^n,p, Z^n,p, ψ^n,p, M^n,p) to BSDE (15) with generator f_n,p. Moreover from Lemma 4, the sequence Y^n,p is bounded since as- sumption (12) holds and the upper bound on Y^n,p does not depend on p. Thus for p large enough, Y^n,p does not depend on p, and is denoted Yⁿ with the same on (Z^n,p, ψ^n,p, M^n,p) = (Zⁿ, ψⁿ, Mⁿ). Now the sequence fn satisfies the assump- tions of the lemmas 2 and 3 with constant independent of n. Thus the sequence (Yⁿ, Uⁿ, Zⁿ, ψⁿ, Mⁿ) is bounded:

sup

n∈N

E Z _T

0

|Y_sⁿ|²+|U_sⁿ|²+|Z_sⁿ|²+ Z

U

|ψⁿ_s(u)|²µ(du)

ds+ [Mⁿ]_T

≤C where U_tⁿ =f_n(t, Y_tⁿ). Therefore there exists a subsequence which converges weakly to (Y, U, Z, ψ, M). We still denote by (Yⁿ, Uⁿ, Zⁿ, ψⁿ, Mⁿ) this subsequence. The Brownian martingaleRT

. Z_sⁿdW_s converges weakly inL²(Ω×[0, T])toRT

. Z_sdW_s(see [30]). The same trick can be applied on the Poisson martingale R_T

.

R

Uψ_sⁿ(u)π(du, ds)e and the orthogonal martingale Mⁿ. Finally we identify U_t and f(t, Y_t) in the same way as in [30].

Finally we remove the condition (12) by a truncation procedure. Once again we obtain a sequence (Yⁿ, Zⁿ, ψⁿ, Mⁿ) which converges inE²(0, T) to the solution(Y, Z, ψ, M) using Lemma 5 (see also proof of Proposition 2).

• Step 3: Using the previous lemma, then we have a mapping Ξ :E²(0, T)→ E²(0, T) which to (U, V, φ, N) ∈ E²(0, T) associates the solution (Y, Z, ψ, M) ∈ E²(0, T) of BSDE (15), and once again it is a contractive mapping with the norm k.k_β with suitable β (same computations as in the proof of Theorem 2.2 in [31]). Hence it has a fixed point(Y, Z, ψ, M), solution of the BSDE (2).

Now we are able to give the main result of this part.

Theorem 1 Under assumptions(Hex)and (H4), there exists a uniqueL²-solution(Y, Z, ψ, M) for the BSDE (2).

Proof. In Proposition 1 the condition (H2) was replaced by (H2’). To obtain the above result we follow the arguments of the proof of Theorem 4.2 in [5] wheref is approximated by a sequence of functions f_n satisfying (H2’) (and the other conditions). Indeed we first assume that ξ and f(t,0,0,0) are bounded. We can construct two sequences f_n and h_n satisfying(Hex)as in [5]. To be more precise, let

• θ_r be a smooth function such that 0 ≤θ_r ≤1, θ_r(y) = 1 if |y| ≤ r and θ_r(y) = 0 as soon as|y| ≥r+ 1;

• ρr(t) = sup_|y|≤r|f(t, y,0,0)−f(t,0,0,0)|;

• for eachn∈N^∗,q_n(z) =zn/(|z| ∨n).

Then

f_n(t, y, z, ψ) = [f(t, y, q_n(z), q_n(ψ))−f(t,0,0,0)] n

ρ_r+1(t)∨n+f(t,0,0,0), hn(t, y, z, ψ) = θr(y) [f(t, y, qn(z), qn(ψ))−f(t,0,0,0)] n

ρ_r+1(t)∨n+f(t,0,0,0).

Note that we also truncate the part onψ inf_n andh_n truncatesf_n for |y| ≥r+ 1.

For fixed r and n, h_n satisfies the conditions of Proposition 1. Hence there exists a unique solution (Yⁿ, Zⁿ, ψⁿ, Mⁿ) in E²(0, T) with generator h_n and from Lemma 4, Yⁿ satisfies the inequalitykYⁿk_∞≤r. Lemma 2 shows that

E Z _T

0

|Z_sⁿ|²ds+ Z _T

0

Z

U

|ψⁿ_s(u)|²µ(du)ds+ [Mⁿ]_T

≤r^′. (16)

Therefore if we have chosen r large enough, (Yⁿ, Zⁿ, ψⁿ, M^m)is solution of the BSDE (2) with generator f_n satisfying (Hex). By Itô’s formula on U =Yⁿ⁺ⁱ−Yⁿ,V =Zⁿ⁺ⁱ−Zⁿ, φ=ψⁿ⁺ⁱ−ψⁿ,N =Mⁿ⁺ⁱ−Mⁿ:

e^4K2t|U_t|²+1 2

Z _T

t

e^4K2s|V_s|²ds+1 2

Z _T

t

Z

U

e^4K2s|φ_s(u)|²µ(du)ds+ Z _T

t

e^4K2sd[N]_s

≤2 Z T

t

e^4K2sU_s(f_n+i(s, Y_sⁿ, Z_sⁿ, ψ_sⁿ)−f_n(s, Y_sⁿ, Z_sⁿ, ψⁿ_s))ds−2 Z T

t

e^4K2sU_sV_sdW_s

−2 Z _T

t

e^4K2sU_s⁻dN_s− Z _T

t

e^4K2s Z

U

|U_s⁻+φ_s(u)|²− |U_s⁻|² e

π(du, ds)

≤4r Z T

t

e^4K2s|fn+i(s, Y_sⁿ, Z_sⁿ, ψ_sⁿ)−fn(s, Y_sⁿ, Z_sⁿ, ψ_sⁿ)|ds−2 Z T

t

e^4K2sUsVsdWs

−2 Z _T

t

e^4K2sU_s⁻dN_s− Z _T

t

e^4K2s Z

U

|U_s⁻+φ_s(u)|²− |U_s⁻|² e

π(du, ds)

since kUk_∞ ≤ 2r. Using the Burkholder-Davis-Gundy (BDG) inequality we get for some constant depending onK and T:

E sup

t∈[0,T]

|U_t|²+ Z _T

0

|V_s|²ds+ Z _T

0

Z

U

|φ_s(u)|²µ(du)ds+ [N]_T

!

≤CrE Z _T

0

|f_n+i(s, Y_sⁿ, Z_sⁿ, ψⁿ_s)−f_n(s, Y_sⁿ, Z_sⁿ, ψ_sⁿ)|ds.

SincekYⁿk_∞≤r, from the definition of f_n we have

|f_n+i(s, Y_sⁿ, Z_sⁿ, ψ_sⁿ)−f_n(s, Y_sⁿ, Z_sⁿ, ψ_sⁿ)|

≤2K|Z_sⁿ|1|Z_sⁿ|>n+ 2Kkψ_sⁿk_L²1kψⁿ_sk_L2>n+ 2K|Z_sⁿ|1ρr+1(s)>n

+2Kkψ_sⁿk_L²1ρr+1(s)>n+ 2ρ_r+1(s)1ρr+1(s)>n.

Since ρr+1 ∈L¹(Ω×[0, T],P⊗m) (Assumption (H2)), and (Zⁿ, ψⁿ)∈H²(0, T)×L²_µ uni- formly w.r.t. n(Inequality (16)), this implies that (Yⁿ, Zⁿ, ψⁿ, M^m) is a Cauchy sequence inE²(0, T).

The general case will be obtained by a truncation procedure on ξ and f(t,0,0,0) and

the inequality of Lemma 5.

3 Existence in L

The following proposition was proved in the Lipschitz case without jumps in [13], Section 5, or in [5] for the Brownian filtration, for any p >1.

Proposition 2 (L^p-estimates, p≥2) We assume that f satisfies (Hex). For p ≥ 2, if we have

E

|ξ|^p+ Z T

0

|f(t,0,0,0)|^pdt

<+∞, (H5)

then the solution (Y, Z, ψ, M) belongs to E^p(0, T). Moreover there exists a constant C de- pending only on K², p and T such that

E

"

sup

t∈[0,T]

|Y_t|^p+ Z T

0

|Z_s|²ds p/2

+ Z T

0

Z

U

|ψ_s(u)|²µ(du)ds p/2

+ [M]^p/2_T

#

≤CE

|ξ|^p+ Z T

0

|f(s,0,0,0)|^pds

. (17)

Proof. Under this condition (H5) on ξ and f(t,0,0,0), we know that there exists a unique solution(Y, Z, ψ, M) which belongs to E²(0, T). We want to show that(Y, Z, ψ, M) in fact belongs toE^p(0, T).

From the proof of Theorem 1 (or Proposition 1), the solution(Y, Z, ψ, M)is obtained as the limit of a sequence(Yⁿ, Zⁿ, ψⁿ, Mⁿ), solution of BSDE (2) but with bounded coefficients ξⁿ and fⁿ(t,0,0,0). We prove that convergence also holds in E^p(0, T) by proving the counterpart of Lemma 5 inE^p. For any (m, n)∈N² we denote

(Y ,b Z,b ψ,b M ,c ξ,bfb) = (Y^m−Yⁿ, Z^m−Zⁿ, ψ^m−ψⁿ, M^m−Mⁿ, ξ^m−ξⁿ, f^m−fⁿ).