L 2 -convergence of Yosida approximation for semi-linear backward stochastic differential equation with jumps in infinite dimension

Purpose – The main motivation of this paper is to present the Yosida approximation of a semi-linear backward stochastic differential equation in infinite dimension. Under suitable assumption and condition, an L 2 -convergence rate is established. Design/methodology/approach – The authors establish a result concerning the L 2 -convergence rate of the solution of backward stochastic differential equation with jumps with respect to the Yosida approximation. Findings – The authors carry out a convergence rate of Yosida approximation to the semi-linear backward stochastic differential equation in infinite dimension. Originality/value – In this paper, the authors present the Yosida approximation of a semi-linear backward stochastic differential equation in infinite dimension. Under suitable assumption and condition, an L 2 -convergence rate is established.


Introduction
Backward stochastic differential equation (BSDE) was performed first by Pardoux and Peng [1] who proved the existence and uniqueness of adapted solutions, under suitable squareintegrability assumptions, on the coefficients and on the terminal condition.Later, several Yosida approximation for semi-linear BSDE authors have been attracted to this area and have provided many applications such as in stochastic games and optimal control [2][3][4], partial differential equations [5] and numerical approximation [6].
The main motivation of this paper is to carry out a convergence rate of the Yosida approximation to the semi-linear backward stochastic differential equation with jumps in infinite dimension.More precisely, let H be a separable Hilbert space with inner product C,D H and H* its dual space.Let V be a uniformly convex Banach space, such that V ⊂ H continuously and densely.For its dual space V*, it follows that H* ⊂ V* continuously and densely.Then by the identification of H and H* via the Riesz isomorphism, we get (V, H, V*) is called a Gelfand triple.
Following [7], we introduce A which is a linear bounded operator such that A: D(A) 5 V → V*, where D(A) 5 {v ∈ V, Av ∈ H}.Using [8], we introduce the Yosida approximation A λ , λ > 0 of A defined as where J: V → V* is the duality mapping defined by Definition 2.1, and J λ : V → V is the resolvent of the operator A is defined by This Yosida approximation is used to approximate the following semi-linear backward stochastic differential equation in infinite dimension: where W is a cylindrical Wiener process, and Ñ is the compensated Poisson random measure.
Using the following family of approximating equations: : where λ > 0, and and A λ is the Yosida approximation, we establish the existence and uniqueness of the solution of (4).Many authors have been devoted to the case of BSDE in infinite dimensional spaces such as [9][10][11].
Hu and Peng [10] proved the existence and the uniqueness of the solution (Y, Z) of this semilinear backward stochastic evolution equations.This kind of equation appears in many topics as those by Bensoussan [12,13] and Hu and Peng [14] for the case with no jumps who have studied the maximum principles for stochastic control systems in infinite dimensional spaces and the theory of optimal control and controllability for stochastic partial differential equations.
Existence and uniqueness of a strong solution of (4) was obtained in Ref. [7] by considering a special case of a backward stochastic evolution equation for Hilbert space valued processes.This, in turn, is studied by taking finite dimensional projections and then taking the limit.This is the Galerkin approximation method which has been used by several authors (See, e.g.Ref. [15]).
The authors in Ref. [21] prove the existence and uniqueness of a solution for a class of backward stochastic differential equations driven by a geometric Brownian motion with a sub-differential operator by means of the Moreau-Yosida approximation method (see Ref. [22] for this used method).Using approximation tools, the authors provide a probabilistic interpretation for the viscosity solutions of a kind of non-linear variational inequalities.
In the same area, the authors in Ref. [23] deal with a class of mean-field backward stochastic differential equations, with sub-differential operator corresponding to a lower semi-continuous convex function.Using Yosida approximation tools, the authors establish the existence and uniqueness of the solution.As an application, they give a probability interpretation for the viscosity solutions of a class of non-local parabolic variational inequalities.
The authors in Ref. [24] propose and analyze multivalued stochastic differential equations (MSDEs) with maximal monotonous operators driven by semimartingales with jumps.They introduce some methods of approximation of solutions of MSDEs based on discretization of processes and Yosida approximation of the monotonous operator.Their paper studies the general problem of stability of solutions of MSDEs with respect to the convergence of driving semimartingales.
Bahlali et al. [25] deal with reflected backward stochastic differential equation (RBSDE) with both monotone and locally monotone coefficient and squared integrable terminal data.Existence and uniqueness of the solution are established with a polynomial growth condition on the coefficient and using Yosida approximation tools.An application to the homogenization of multivalued partial differential equations is given by the authors.The aim of our paper differs from the one proposed in Ref. [26], as it concentrates on BSDEs instead of SDEs.Additionally, it differs from the approach described in Ref. [7] by integrating the idea of L 2 -convergence of Yosida approximation.This integration offers a possible technique for solving multivalued differential equations.
This paper is composed of four sections.Section 2 introduces some notations, the Yosida approximation approach and preliminaries results.Section 3 establishes a result concerning the L 2 -convergence rate of the solution of backward stochastic differential equation with jumps with respect to the Yosida approximation.In Section 4, we carry out a convergence rate of the Yosida approximation to the semi-linear backward stochastic differential equation in infinite dimension.

Preliminaries and notations
Let ðΩ; F ; PÞ be a probability space with filtration ðF t Þ t∈½0;T ∈ F. Let Ξ, H be two separable Hilbert spaces, and H* be the dual space of H. Let V be a Banach space dense in H. Let us assume that V is uniformly convex with uniformly convex dual V*.It follows that H* ⊂ V* continuously and densely.Then, by the identification of H and H* via the Riesz isomorphism, we get V ⊂ H ⊂ V * : The Milman-Pettis theorem (see, e.g.Yosida [ [27], p. 127]) states that every uniformly convex Banach space is reflexive.So, V is a reflexive Banach space.

Yosida approximation for semi-linear BSDE
Following [28], we introduce a cylindrical Wiener process in Ξ as a family (W(t), t ≥ 0), of linear applications Ξ → L 2 (Ω) such that: (1) For every h ∈ Ξ, {W(t)h, t ≥ 0} is a real (continuous) Wiener process, Let ðE; BðEÞÞ be a measurable space, where E is a topological vector space.Furthermore, let ξ(t) be a L evy process on E and be denoted by ν(dx), the L evy measure of ξ.Denote by L 2 (ν) the L 2 -space of square integrable H À valued measurable functions associated with ν.
We denote by P the predictable σ À field on Ω 3 [0, T].Introduce now the following spaces: (1) L 2 (0, T, H): the set of all F t − progressively measurable processes takes its values in H, such that (2) L 2 (Ξ, H): the set of the Hilbert-Schmidt operators from Ξ to H, that is, where fe n g ∞ n¼1 is an orthonormal basis on Ξ.The set Moreover, beside the same hypotheses on the cylindrical Wiener process, we have: (1) A positive number T > 0; (2) A map f: (4) A bounded linear operator A: D(A) 5 V → V*, where D(A) 5 {v ∈ V, Av ∈ H}.We assume that the operator A is monotone, meaning: Now, we assume the following useful hypothesis denoted by Hyp.1: (2) There exists a constant C > 0, such that P almost surely for almost every t ∈ [0, T], the following holds for all In most cases, the duality mapping defined here is multivalued.
Definition 2.1.The duality mapping J: V → V* is defined by: Under hypotheses of V and V*, we get the following result: Theorem 2.2.[20] Let V be a Banach space.If V* is strictly convex, then the duality mapping J: V → V* is single-valued.
For the detailed proof, see Theorem 1.2 in Ref. [8].
We will now provide an approximation of the operator A, as mentioned in Ref. [8].
Definition 2.4.For every x ∈ V and λ > 0, the Yosida approximation of A is defined by the operator A λ : V → V* as where the resolvent J λ : V → V of the operator A is defined by J λ x 5 x λ , with x λ as a unique solution to the equation: The uniqueness of x λ was proved by [20] [Proposition 3.17.p. 36].According to [8] [Proposition 1.3],A λ is single-valued, monotone, bounded on bounded subsets and semicontinuous from V to V*.The resolvent can be written as Lemma 2.5.Equation ( 8) can be reformulated as: Proof.Let x ∈ V and J λ (x) be the resolvents of the operator A defined by equation (10).By the definition of the Yosida approximation and the homogeneity of J À1 (see Ref. [20]), Equation ( 8) can be written as Yosida approximation for semi-linear BSDE J λ ðxÞ ¼ x À λJ −1 ðA λ ðxÞÞ: Using the fact that A λ (x) 5 A(J λ (x)) for all x ∈ V ([ [20], Proposition 3.19]) and inserting this into the resolvent equation ( 9), we obtain A λ (x) 5 A(x À λJ À1 (A λ (x))) or equivalently, x 5 . Since A λ is single-valued, we conclude (11).

Yosida approximation
Let H be a separable Hilbert space and V a Banach space such that the space V ⊂ H is reflexive and dense in H.We identify H with its dual space H*, and V with its dual space V*.Then, we get We denote by j$j V , j$j V* , j$j H , the norms in V, V* and H, respectively, and by C, D the duality product between V and V*.We introduce the following application: which verifies the following coercivity condition (L1): In this section, we are interested in the Yosida approximation of the following semi-linear backward stochastic differential equation in infinite dimension: Let us consider the family of approximating equations of ( 12) Remark 3.1.Note that, for all λ > 0, the operator A λ being linear and bounded [[8], Proposition 2.2], it is checked by the standard PicardÀLindelof iteration methods [7] that the triplet (Y λ , Z λ , Q λ ) is a classical solution of ( 13), and it verifies for all t ∈ [0, T], that The following result establishes the existence and the uniqueness of the solution of (12).

AJMS
The following results will be used to prove our main result about the L 2 convergence rate.

Yosida approximation for semi-linear BSDE
where By the Gronwall lemma, we finally obtain the expression (15).
The following remark plays a fundamental role in the convergence rate of Yosida approximation.for all x ∈ D(A) on [0, T] and by using the fact that D(A) 5 V and we get under Condition (L1) and Hyp.1, we then obtain by applying lemma 3.4:

Convergence of Yosida approximation
In this section, we prove a convergence rate of Yosida approximation to the following semilinear backward stochastic differential equation in infinite dimension: Proposition 4.1.Let Y λ be the solution to the backward stochastic differential equation (12), and assume that Hyp.1 holds.Let λ, μ > 0, then there exists D > 0, such that: Proof.Let us denote by Y λ t and Y μ t two Yosida approximation to by Itô formula, then the expectation, we get By definition of A λ and the bijectivity of J λ , we have I 5 J λ þ J À1 (λA λ ).Hence: So by using Lemma 2.5, we obtain A λ 5 AJ λ and A μ 5 AJ μ .Then the monotonicity of A (5) and the fact that J À1 is the duality map from V* to V** 5 V, the first aforementioned term is positive, so we get where we have used the elementary inequality 2ab ≤ a 2 þ b 2 .Here, by applying the expectation and Lipschitz condition Hyp.1 of f, we get Then, we obtain where Yosida approximation for semi-linear BSDE Using Gronwall lemma, this shows that EjY λ t − Y μ t j 2 H ≤ B λ;μ e C α ðT−tÞ , which plugged in the inequality (19) provides where By subtraction, we have: For α larger than (1 þ C 2 (T À t))C, this provides that there exists D > 0, such that By using the same idea for the jump part and plugging in (19), we deduce that Using that A λ and A μ , we verify the boundedness condition introduced in Remark 3.5, and the result holds.
The following theorem shows that the limit (Y, Z, Q) is a solution of equation ( 12).
Proof.For the detailed proof, we refer to [28] where the resolvent J λ of A is defined on H by A satisfies (L1).
Proof.For more details, we refer to [ [28], p. 59].Proof.For a detailed proof, we refer to [28] a stationary Poisson point process on E with characteristic measure ν.Denote by N(dt, dx) the Poisson counting measure associated with the L evy process, Nðt; AÞ ¼ P s∈Dp s≤t I A ðpðsÞÞ.Denote by Ñ ðdt; dxÞ ¼ N ðdt; dxÞ − dtνðdxÞ the compensated Poisson random measure.The filtration is defined as follows

2 L 2 Example 4 . 5 .
ðνÞ ds ≤ 2Dλ: Let an open set Λ ⊂ R d , and denote by C ∞ 0 ðΛÞ the set of all infinitely differentiable real valued functions defined on Λ with compact support.For u ∈ C ∞