Idris Kharroubi

Ceremade

Nicolas Langrené

Huyên Pham

Discrete time approximation of fully nonlinear HJB equations via BSDEs with nonpositive jumps

Keywords: Discrete time approximation, Hamilton-Jacobi-Bellman equation, nonlinear degenerate PDE, optimal control, backward stochastic differential equations. MSC Classification: 65C99, 60J75, 49L25

.

Introduction

Let us consider the fully nonlinear generalized Hamilton-Jacobi-Bellman (HJB) equation:

   ∂v ∂t + sup a∈A b(x, a).D x v + 1 2 tr(σσ ⊺ (x, a)D 2 x v) + f (x, a, v, σ ⊺ (x, a)D x v) = 0, on [0, T ) × R d , v(T, x) = g, on R d .
(1.1) In the particular case where f (x, a) does not depend on v and D x v, this partial differential equation (PDE) is the dynamic programming equation for the stochastic control problem:

v(t, x) = sup α E T t f (X α s , α s )ds + g(X α T ) X α t = x , (1.2)
with controlled diffusion in R d :

dX α t = b(X α t , α t )dt + σ(X α t , α t )dW t ,
and where α is an adapted control process valued in a compact space A of R q . Numerical methods for parabolic partial differential equations (PDEs) are largely developed in the literature, but remain a big challenge for fully nonlinear PDEs, like the HJB equation (1.1), especially in high dimensional cases. We refer to the recent paper [START_REF] Fahim | A Probabilistic Numerical Scheme for Fully Nonlinear PDEs[END_REF] for a review of some deterministic and probabilistic approaches.

In this paper, we propose a new probabilistic numerical scheme for HJB equation, relying on the following Feynman-Kac formula for HJB equation obtained by randomization of the control process α. We consider the minimal solution (Y, Z, U, K) to the backward stochastic differential equation (BSDE) with nonpositive jumps:

     Y t = g(X T ) + T t f (X s , I s , Y s , Z s )ds + K T -K t - T t Z s dW s - T t
A U s (a)μ(ds, da), 0 ≤ t ≤ T, U t (a) ≤ 0, (1.3) with a forward Markov regime-switching diffusion process (X, I) valued in R d × A given by:

X t = X 0 + t 0 b(X s , I s )ds + t 0 σ(X s , I s )dW s I t = I 0 + (0,t] A (a -I s -)µ(ds, da).
Here W is a standard Brownian motion, µ(dt, da) is a Poisson random measure on [0, ∞)×A with finite intensity measure λ(da) of full topological support on A, and compensated measure μ(dt, da) = µ(dt, da) -λ(da)dt. Assumptions on the coefficients b, σ, f, g will be detailed in the next section, but we emphasize the important point that no degeneracy condition on the controlled diffusion coefficient σ is imposed. It is proved in [START_REF] Kharroubi | Feynman-Kac representation for Hamilton-Jacobi-Bellman IPDE[END_REF] that the minimal solution to this class of BSDE is related to the HJB equation (1.1) through the relation Y t = v(t, X t ).

The purpose of this paper is to provide and analyze a discrete-time approximation scheme for the minimal solution to (1.3), and thus an approximation scheme for the HJB equation. In the non-constrained jump case, approximations schemes for BSDE have been studied in the papers [START_REF] Gobet | Rate of convergence of empirical regression method for solving generalized BSDE[END_REF], [START_REF] Bouchard | Discrete time approximation of decoupled FBSDE with jumps[END_REF], which extended works in [START_REF] Bouchard | Discrete time approximation and Monte-Carlo simulation of backward stochastic differential equations[END_REF], [START_REF] Zhang | A numerical scheme for BSDEs[END_REF] for BSDEs in a Brownian framework. The issue is now to deal with the nonpositive jump constraint in (1.3), and we propose a discrete time approximation scheme of the form:

               Ȳ π T = Ȳπ T = g( Xπ T ) Zπ t k = E Ȳ π t k+1 Wt k+1 -Wt k t k+1 -t k F t k Ȳπ t k = E Ȳ π t k+1 F t k + (t k+1 -t k ) f ( Xπ t k , I t k , Ȳπ t k , Zπ t k ) Ȳ π t k = ess sup a∈A E Ȳπ t k F t k , I t k = a , k = 0, . . . , n -1, (1.4)
where π = {t 0 = 0 < . . . < t k < . . . < t n = T } is a partition of the time interval [0, T ], with modulus |π|, and Xπ is the Euler scheme of X (notice that I is perfectly simulatable once we know how to simulate the distribution λ(da)/ A λ(da) of the jump marks). The interpretation of this scheme is the following. The first three lines in (1.4) correspond to the standard scheme ( Ȳπ , Zπ ) for a discretization of a BSDE with jumps (see [START_REF] Bouchard | Discrete time approximation of decoupled FBSDE with jumps[END_REF]), where we omit here the computation of the jump component. The last line in (1.4) for computing the approximation Ȳ π of the minimal solution Y corresponds precisely to the minimality condition for the nonpositive jump constraint and should be understood as follows. By the Markov property of the forward process (X, I), the solution (Y, Z, U ) to the BSDE with jumps (without constraint) is in the form Y t = ϑ(t, X t , I t ) for some deterministic function ϑ. Assuming that ϑ is a continuous function, the jump component of the BSDE, which is induced by a jump of the forward component I, is equal to U t (a) = ϑ(t, X t , a)-ϑ(t, X t , I t -). Therefore, the nonpositive jump constraint means that: ϑ(t, X t , I t -) ≥ ess sup a∈A ϑ(t, X t , a).

The minimality condition is thus written as:

Y t = v(t, X t ) = ess sup a∈A ϑ(t, X t , a) = ess sup a∈A E[Y t |X t , I t = a],
whose discrete time version is the last line in scheme (1.4).

In this work, we mainly consider the case where f (x, a, y) does not depend on z, and our aim is to analyze the discrete time approximation error on Y , where we split the error between the positive and negative parts:

Err π + (Y ) := max k≤n-1 E Y t k -Ȳ π t k 2 + 1 2 , Err π -(Y ) := max k≤n-1 E Y t k -Ȳ π t k 2 - 1 2 .
We do not study directly the error on Z, and instead focus on the approximation of an optimal control for the HJB equation, which is more relevant in practice. It appears that the maximization step in the scheme (1.4) provides a control in feedback form {â(t k , Xπ t k ), k ≤ n-1}, which approximates the optimal control with an estimated error bound. The analysis of the error on Y proceeds as follows. We first introduce the solution (Y π , Y π , Z π , U π ) of a discretely jump-constrained BSDE. This corresponds formally to BSDEs for which the nonpositive jump constraint operates only a finite set of times, and should be viewed as the analog of discretely reflected BSDEs defined in [START_REF] Bally | Error analysis of the quantization algorithm for obstacle problems[END_REF] and [START_REF] Bouchard | Discrete time approximation for continuously and discretely reflected BSDEs[END_REF] in the context of the approximation for reflected BSDEs. By combining BSDE methods and PDE approach with comparison principles, and further with the shaking coefficients method of Krylov [START_REF] Krylov | On the rate of convergence of finite difference approximations for Bellman's equations with variable coefficients[END_REF] and Barles, Jacobsen [START_REF] Barles | Error bounds for monotone approximation schemes for parabolic Hamilton-Jacobi-Bellman equations[END_REF], we prove the monotone convergence of this discretely jump-constrained BSDE towards the minimal solution to the BSDE with nonpositive jump constraint, and obtained a convergence rate without any ellipticity condition on the diffusion coefficient σ. We next focus on the approximation error between the discrete time scheme in (1.4) and the discretely jump-constrained BSDE. The standard argument for studying rate of convergence of such error consists in getting an estimate of the error at time

t k : E[|Y π t k -Ȳ π t k | 2
] in function of the same estimate at time t k+1 , and then conclude by induction together with classical estimates for the forward Euler scheme. However, due to the supremum in the conditional expectation in the scheme (1.4) for passing from Ȳπ to Ȳ π , which is a nonlinear operation violating the law of iterated conditional expectations, such argument does not work anymore. Instead, we consider the auxiliary error control at time t k : for the global error Err π (Y ) = Err π + (Y ) + Err π -(Y ). Moreover, in the case where f (x, a) does not depend on y (i.e. the case of standard HJB equation and stochastic control problem), we obtain a better rate of order |π| 1 6 by relying on a stochastic control representation of the discretely jump-constrained BSDE, and by using a convergence rate result in [START_REF] Krylov | Approximating value functions for controlled degenerate diffusion processes by using piece-wise constant policies[END_REF] for the approximation of controlled diffusion by means of piece-wise constant policies. Anyway, our result improves the convergence rate of the mixed Monte-Carlo finite difference scheme proposed in [START_REF] Fahim | A Probabilistic Numerical Scheme for Fully Nonlinear PDEs[END_REF], where the authors obtained a rate |π| We conclude this introduction by pointing out that the above discrete time scheme is not yet directly implemented in practice, and requires the estimation and computation of the conditional expectations together with the supremum. Actually, simulation-regression methods on basis functions defined on R d × A appear to be very efficient, and provide approximate optimal controls in feedback forms via the maximization operation in the last step of the scheme (1.4). We postpone this analysis and illustrations with several numerical tests arising in superreplication of options under uncertain volatility and correlation in a companion paper [START_REF] Kharroubi | Numerical algorithm for fully nonlinear HJB equations: an approach by control randomization[END_REF]. Notice that since it relies on the simulation of the forward process (X, I), our scheme does not suffer the curse of dimensionality encountered in finite difference scheme or controlled Markov chains methods (see [START_REF] Kushner | Numerical methods for stochastic control problems in continuous time[END_REF], [START_REF] Bonnans | Consistency of generalized finite difference schemes for the stochastic HJB equation[END_REF]), and takes advantage of the highdimensional properties of Monte-Carlo methods.

E π k (Y) := E ess sup a∈A E t 1 ,a . . . ess sup a∈A E t k ,a |Y π t k -Ȳπ t k | 2 . . . ,
The remainder of the paper is organized as follows. In Section 2, we state some useful auxiliary error estimate for the Euler scheme of the regime switching forward process. We introduce in Section 3 discretely jump-constrained BSDE and relate it to a system of integropartial differential equations. Section 4 is devoted to the convergence of discretely jumpconstrained BSDE to the minimal solution of BSDE with nonpositive jumps. We provide in Section 5 the approximation error for our discrete time scheme, and as a byproduct an estimate for the approximate optimal control in the case of classical HJB equation associated to stochastic control problem.

The forward regime switching process

Let (Ω, F, P) be a probability space supporting d-dimensional Brownian motion W , and a Poisson random measure µ(dt, da) with intensity measure λ(da)dt on [0, ∞) × A, where A is a compact set of R q , endowed with its Borel tribe B(A), and λ is a finite measure on (A, B(A)) with full topological support. We denote by F = (F t ) t≥0 the completion of the natural filtration generated by (W, µ), and by P the σ-algebra of F-predictable subsets of Ω × R + .

We fix a finite time horizon T > 0, and consider the solution (X, I) on [0, T ] of the regime-switching diffusion model:

X t = X 0 + t 0 b(X s , I s )ds + t 0 σ(X s , I s )dW s I t = I 0 + (0,t] A (a -I s -)µ(ds, da), (2.1)
where (X

0 , I 0 ) ∈ R d × A, b : R d × A → R d and σ : R d × A → R d×d ,

are measurable functions, satisfying the Lipschitz condition:

(H1) There exists a constant L 1 such that

|b(x, a) -b(x ′ , a ′ )| + |σ(x, a) -σ(x ′ , a ′ )| ≤ L 1 |x -x ′ | + |a -a ′ | , for all x, x ′ ∈ R d and a, a ′ ∈ A.
The assumption (H1) stands in force throughout the paper, and in this section, we shall denote by C 1 a generic positive constant which depends only on L 1 , T , (X 0 , I 0 ) and λ(A) < ∞, and may vary from lines to lines. Under (H1), we have the existence and uniqueness of a solution to (2.1), and in the sequel, we shall denote by (X t,x,a , I t,a ) the solution to (2.1) starting from (x, a) at time t.

Remark 2.1 We do not make any ellipticity assumption on σ. In particular, some lines and columns of σ may be equal to zero, and so there is no loss of generality by considering that the dimension d of X and W are equal. ✷

We first study the discrete-time approximation of the forward process. Denoting by (T n , ι n ) n the jump times and marks associated to µ, we observe that I is explicitly written as:

I t = I 0 1 [0,T 1 ) (t) + n≥1 ι n 1 [Tn,T n+1 ) (t), 0 ≤ t ≤ T,
where the jump times (T n ) n evolve according to a Poisson distribution of parameter λ := A λ(da) < ∞, and the i.i.d. marks (ι n ) n follow a probability distribution λ(da) := λ(da)/λ. Assuming that one can simulate the probability distribution λ, we then see that the pure jump process I is perfectly simulated. Given a partition π = {t 0 = 0 < . . . < t k < . . . t n = T } of [0, T ], we shall use the natural Euler scheme Xπ for X, defined by:

Xπ 0 = X 0 Xπ t k+1 = Xπ t k + b( Xπ t k , I t k )(t k+1 -t k ) + σ( Xπ t k , I t k )(W t k+1 -W t k ),
for k = 0, . . . , n-1. We denote as usual by |π| = max k≤n-1 (t k+1 -t k ) the modulus of π, and assume that n|π| is bounded by a constant independent of n, which holds for instance when the grid is regular, i.e. (t k+1 -t k ) = |π| for all k ≤ n-1. We also define the continuous-time version of Xπ by setting:

Xπ t = Xπ t k + b( Xπ t k , I t k )(t -t k ) + σ( Xπ t k , I t k )(W t -W t k ), t ∈ [t k , t k+1 ], k < n.
By standard arguments, see e.g. [START_REF] Kloeden | Numerical solution of stochastic differential equations[END_REF], one can obtain under (H1) the L 2 -error estimate for the above Euler scheme:

E sup t∈[t k ,t k+1 ] X t -Xπ t k 2 ≤ C 1 |π|, k < n.
For our purpose, we shall need a stronger result, and introduce the following error control for the Euler scheme:

E π k (X) := E ess sup a∈A E t 1 ,a . . . ess sup a∈A E t k ,a sup t∈[t k ,t k+1 ] |X t -Xπ t k | 2 . . . , (2.2)
X t -Xπ t k 2 ≤ E π k (X), k < n. Lemma 2.1 We have max k<n E π k (X) ≤ C 1 |π|.
Proof. From the definition of the Euler scheme, and under the growth linear condition in (H1), we easily see that

E t k Xπ t k+1 2 ≤ C 1 1 + Xπ t k 2 , k < n. (2.3)
From the definition of the continuous-time Euler scheme, and by Burkholder-Davis-Gundy inequality, it is also clear that

E t k sup t∈[t k ,t k+1 ] Xπ t -Xπ t k 2 ≤ C 1 (1 + Xπ t k 2 |π|, k < n. (2.4)
We also have the standard estimate for the pure jump process I (recall that A is assumed to be compact and λ(A) < ∞):

E t k sup t∈[t k ,t k+1 ] I s -I t k 2 ≤ C 1 |π|. (2.5)
Let us denote by ∆X t = X t -Xπ t , and apply Itô's formula to |∆X t | 2 so that for all t ∈ [t k , t k+1 ]:

|∆X t | 2 = |∆X t k | 2 + t t k 2 b(X s , I s ) -b( Xπ t k , I t k ) .∆X s + σ(X s , I s ) -σ( Xπ t k , I t k ) 2 ds + 2 t t k (∆X s ) ′ σ(X s , I s ) -σ( Xπ t k , I t k ) dW s ≤ |∆X t k | 2 + C 1 t t k |∆X s | 2 + | Xπ s -Xπ t k | 2 + |I s -I t k | 2 ds + 2 t t k (∆X s ) ′ σ(X s , I s ) -σ( Xπ t k , I t k ) dW s ,
from the Lipschitz condition on b, σ in (H1). By taking conditional expectation in the above inequality, we then get:

E t k |∆X t | 2 ≤ |∆X t k | 2 + C 1 t t k E t k |∆X s | 2 + | Xπ s -Xπ t k | 2 + |I s -I t k | 2 ds ≤ |∆X t k | 2 + C 1 (1 + Xπ t k 2 |π| 2 + C 1 t t k E t k |∆X s | 2 ds, t ∈ [t k , t k+1 ],
by (2.4)-(2.5). From Gronwall's lemma, we thus deduce that

E t k |∆X t k+1 | 2 ≤ e C 1 |π| |∆X t k | 2 + C 1 (1 + Xπ t k 2 |π| 2 , k < n. (2.6)
Since the right hand side of (2.6) does not depend on I t k , this shows that ess sup

a∈A E t k ,a |∆X t k+1 | 2 ≤ e C 1 |π| |∆X t k | 2 + C 1 (1 + Xπ t k 2 |π| 2 .
By taking conditional expectation w.r.t. F t k-1 in the above inequality, using again estimate (2.6) together with (2.3) at step k-1, and iterating this backward procedure until the initial time t 0 = 0, we obtain:

E ess sup a∈A E t 1 ,a . . . ess sup a∈A E t k ,a |∆X t k+1 | 2 . . . ≤ e C 1 n|π| |∆X 0 | 2 + C 1 (1 + |X 0 | 2 )|π| 2 e C 1 n|π| -1 e C 1 |π| -1 ≤ C 1 |π|, (2.7)
since ∆X 0 = 0 and n|π| is bounded. Moreover, the process X satisfies the standard conditional estimate similarly as for the Euler scheme:

E t k X t k+1 2 ≤ C 1 1 + X t k 2 , E t k sup t∈[t k ,t k+1 ] X t -X t k 2 ≤ C 1 (1 + X t k 2 |π|, k < n,
from which we deduce by backward induction on the conditional expectations: Given the forward regime switching process (X, I) defined in the previous section, we consider the minimal quadruple solution (Y, Z, U, K) to the BSDE with nonpositive jumps:

E
     Y t = g(X T ) + T t f (X s , I s , Y s , Z s )ds + K T -K t - T t Z s dW s - T t A U s (a)μ(ds, da), 0 ≤ t ≤ T. U t (a) ≤ 0, (3.1) By solution to (3.1), we mean a quadruple (Y, Z, U, K) ∈ S 2 × L 2 (W ) × L 2 (μ) × K 2 , where S 2 is the space of càd-làg or càg-làd F-progressively measurable processes Y satisfying Y 2 := E[sup t∈[0,T ] |Y t | 2 ] < ∞, L 2 (W ) is the space of R d -valued P-measurable processes such that Z 2 L 2 (W ) := E[ T 0 |Z t | 2 dt] < ∞, L 2 (μ) is the space of real-valued P ⊗B(A)-measurable processes U such that U 2 L 2 (μ) := E[ T 0 A |U t (a)| 2 λ(da) dt] < ∞,
and K 2 is the subspace of S 2 consisting of nondecreasing predictable processes such that K 0 = 0, P-a.s., and the equation in (3.1) holds P-a.s., while the nonpositive jump constraint holds on Ω × [0, T ] × A a.e. with respect to the measure dP ⊗ dt ⊗ λ(da). By minimal solution to the BSDE (1.3), we mean a quadruple solution (Y,

Z, U, K) ∈ S 2 × L 2 (W ) × L 2 (μ) × K 2 such that for any other solution (Y ′ , Z ′ , U ′ , K ′ ) to the same BSDE, we have P-a.s.: Y t ≤ Y ′ t , t ∈ [0, T ].
In the rest of this paper, we shall make the standing Lipschitz assumption on the functions f :

R d × A × R × R d → R and g : R d → R. (H2) There exists a constant L 2 such that |f (x, a, y, z) -f (x ′ , a ′ , y ′ , z ′ )| + |g(x) -g(x ′ )| ≤ L 2 |x -x ′ | + |a -a ′ | + |y -y ′ | + |z -z ′ | , for all x, x ′ ∈ R d , y, y ′ ∈ R, z, z ′ ∈ R d , a, a ′ ∈ A.
In the sequel, we shall denote by C a generic positive constant which depends only on L 1 , L 2 , T , (X 0 , I 0 ) and λ(A) < ∞, and may vary from lines to lines. Under (H1)-(H2), it is proved in [START_REF] Kharroubi | Feynman-Kac representation for Hamilton-Jacobi-Bellman IPDE[END_REF] the existence and uniqueness of a minimal solution (Y, Z, U, K) to (3.1). Moreover, the minimal solution Y is in the form

Y t = v(t, X t ), 0 ≤ t ≤ T, (3.2)
where v : [0, T ] × R d → R is a viscosity solution with linear growth to the fully nonlinear HJB type equation:

-

sup a∈A L a v + f (x, a, v, σ ⊺ (x, a)D x v) = 0, on [0, T ) × R d , v(T, x) = g, on R d , (3.3)
where

L a v = ∂v ∂t + b(x, a).D x v + 1 2 tr(σσ ⊺ (x, a)D 2 x v).
We shall make the standing assumption that comparison principle holds for (3.3).

(HC) Let w (resp. w) be a lower-semicontinuous (resp. upper-semicontinuous) viscosity supersolution (resp. subsolution) with linear growth condition to (3.3). Then, w ≥ w.

When f does not depend on y, z, i.e. (3.3) is the usual HJB equation for a stochastic control problem, Assumption (HC) holds true, see [START_REF] Fleming | Controlled Markov Processes and Viscosity Solutions[END_REF] or [START_REF] Pham | Continuous-time stochastic control and optimization with financial applications[END_REF]. In the general case, we refer to [START_REF] Crandall | User's guide to viscosity solutions of second order partial differential equations[END_REF] for sufficient conditions to comparison principles. Under (HC), the function v in (3.2) is the unique viscosity solution to (3.3), and is in particular continuous. Actually, we have the standard Hölder and Lipschitz property (see Appendix in [START_REF] Krylov | On the rate of convergence of finite difference approximations for Bellman's equations with variable coefficients[END_REF] or [START_REF] Barles | Error bounds for monotone approximation schemes for parabolic Hamilton-Jacobi-Bellman equations[END_REF]):

|v(t, x) -v(t ′ , x ′ )| ≤ C |t -t ′ | 1 2 + |x -x ′ | , (t, t ′ ) ∈ [0, T ], x, x ′ ∈ R d . (3.4)
This implies that the process Y is continuous, and thus the jump component U = 0. In the sequel, we shall focus on the approximation of the remaining components Y and Z of the minimal solution to (3.1).

We introduce in this section discretely jump-constrained BSDE. The nonpositive jump constraint operates only at the times of the grid π = {t 0 = 0 < t 1 < . . . < t n = T } of [0, T ], and we look for a quadruple

(Y π , Y π , Z π , U π ) ∈ S 2 × S 2 × L 2 (W ) × L 2 (μ) satisfying: Y π T = Y π T = g(X T ) (3.5)
and

Y π t = Y π t k+1 + t k+1 t f (X s , I s , Y π s , Z π s )ds (3.6) - t k+1 t Z π s dW s - t k+1 t A U π s (a)μ(ds, da) , Y π t = Y π t 1 (t k ,t k+1 ) (t) + ess sup a∈A E Y π t X t , I t = a 1 {t k } (t) , (3.7)
for all t ∈ [t k , t k+1 ) and all 0 ≤ k ≤ n -1.

Notice that at each time t k of the grid, the condition is not known a priori to be square integrable since it involves a supremum over A, and the well-posedness of the BSDE (3.5)-(3.6)-(3.7) is not a direct and standard issue. We shall use a PDE approach for proving the existence and uniqueness of a solution. Let us consider the system of integropartial differential equations (IPDEs) for the functions v π and ϑ π defined recursively on [0, T ] × R d × A by:

• A terminal condition for v π and ϑ π :

v π (T, x, a) = ϑ π (T, x, a) = g(x) , (x, a) ∈ R d × A , (3.8) • A sequence of IPDEs for ϑ π      -L a ϑ π -f x, a, ϑ π , σ ⊺ (x, a)D x ϑ π -A ϑ π (t, x, a ′ ) -ϑ π (t, x, a) λ(da ′ ) = 0, (t, x, a) ∈ [t k , t k+1 ) × R d × A, ϑ π (t - k+1 , x, a) = sup a ′ ∈A ϑ π (t k+1 , x, a ′ ) (x, a) ∈ R d × A (3.9) for k = 0 . . . , n -1,
• the relation between v π and ϑ π : For any L-Lipschitz continuous function ϕ on R d × A, and k ≤ n -1, we denote:

v π (t, x, a) = ϑ π (t, x, a)1 (t k ,t k+1 ) (t) + sup a ′ ∈A ϑ π (t, x, a ′ )1 {t k } (t) , (3.10
T k π [ϕ](t, x, a) := w(t, x, a), (t, x, a) ∈ [t k , t k+1 ) × R d × A, (3.11)
where w is the unique continuous viscosity solution on [t k , t k+1 ] × R d × A with linear growth condition in x to the integro partial differential equation (IPDE):

     -L a w -f (x, a, w, σ ⊺ D x w) -A w(t, x, a ′ ) -w(t, x, a) λ(da ′ ) = 0, (t, x, a) ∈ [t k , t k+1 ) × R d × A, w(t - k+1 , x, a) = ϕ(x, a), (x, a) ∈ R d × A , (3.12)
and we extend by continuity T k π [ϕ](t k+1 , x, a) = ϕ(x, a). The existence and uniqueness of such a solution w to the semi linear IPDE (3.12), and its nonlinear Feynman-Kac representation in terms of BSDE with jumps, is obtained e.g. from Theorems 3.4 and 3.5 in [START_REF] Barles | BSDEs and integral-partial differential equations[END_REF]. Lemma 3.1 There exists a constant C such that for any L-Lipschitz continuous function ϕ on R d × A, and k ≤ n -1, we have

|T k π [ϕ](t, x, a) -T k π [ϕ](t, x ′ , a ′ )| ≤ max(L, 1) 1 + |π|e C|π| (|x -x ′ | + |a -a ′ |) , for all t ∈ [t k , t k+1 ), and (x, a), (x ′ , a ′ ) ∈ R d × A. Proof. Fix t ∈ [t k , t k+1 ), k ≤ n-1, (x, a), (x ′ , a ′ ) ∈ R d ×A, and ϕ an L-Lipschitz continuous function on R d × A. Let (Y ϕ , Z ϕ , U ϕ ) and (Y ϕ, ′ , Z ϕ, ′ , U ϕ, ′ ) be the solutions on [t, t k+1 ] to the BSDEs Y ϕ s = ϕ(X t,x,a t k+1 , I t,a t k+1 ) + t k+1 s f (X t,x,a r , I t,a r , Y ϕ r , Z ϕ r )dr - t k+1 s Z ϕ r dW r - t k+1 s A U ϕ r (e)μ(dr, de), t ≤ s ≤ t k+1 , Y ϕ, ′ s = ϕ(X t,x ′ ,a ′ t k+1 , I t,a ′ t k+1 ) + t k+1 s f (X t,x ′ ,a ′ r , I t,a ′ r , Y ϕ, ′ r , Z ϕ, ′ r )dr - t k+1 s Z ϕ, ′ r dW r - t k+1 s A U ϕ, ′ r (e)μ(dr, de), t ≤ s ≤ t k+1
From Theorems 3.4 and 3.5 in [START_REF] Barles | BSDEs and integral-partial differential equations[END_REF], we have the identification:

Y ϕ t = T k π [ϕ](t, x, a) and Y ϕ, ′ t = T k π [ϕ](t, x ′ , a ′ ) . (3.13)
We now estimate the difference between the processes Y ϕ and Y ϕ, ′ , and set

δY ϕ = Y ϕ -Y ϕ, ′ , δZ ϕ = Z ϕ -Z ϕ, ′ , δX = X t,x,a -X t,
x ′ ,a ′ , δI = I t,a -I t,a ′ . By Itô's formula, the Lipschitz condition of f and ϕ, and Young inequality, we have

E |δY ϕ s | 2 + E t k+1 s |δZ ϕ s | 2 ds ≤ L 2 E |δX T | 2 + |δI T | 2 + C t k+1 s E |δY ϕ r | 2 dr + 1 2 E t k+1 s |δX r | 2 + |δI r | 2 + |δZ ϕ r | 2 dr
E |δX r | 2 + |δI r | 2 ≤ e C|π| |x -x ′ | 2 + |a -a ′ | 2 ,
and thus:

E |δY ϕ s | 2 ≤ (L 2 + |π|)e C|π| |x -x ′ | 2 + |a -a ′ | 2 + C t k+1 s E |δY ϕ r | 2 dr , for all s ∈ [t, t k+1 ]. By Gronwall's Lemma, this yields sup s∈[t,t k+1 ] E |δY ϕ s | 2 ≤ (L 2 + |π|)e 2C|π| |x -x ′ | 2 + |a -a ′ | 2 ,
which proves the required result from the identification (3.13):

|T k π [ϕ](t, x, a) -T k π [ϕ](t, x ′ , a ′ )| ≤ L 2 + |π|e C|π| (|x -x ′ | + |a -a ′ |) ≤ max(L, 1) 1 + |π|e C|π| (|x -x ′ | + |a -a ′ |).

✷ Proposition 3.1

There exists a unique viscosity solution ϑ π with linear growth condition to the IPDE (3.8)-(3.9), and this solution satisfies:

|ϑ π (t, x, a) -ϑ π (t, x ′ , a ′ )| ≤ max(L 2 , 1) e 2C|π| (1 + |π|) n-k |x -x ′ | + |a -a ′ | , (3.14)
for all k = 0, . . . , n -1, t ∈ [t k , t k+1 ), (x, a), (x ′ , a ′ ) ∈ R d × A.
Proof. We prove by a backward induction on k that the IPDE (3.8)-(3.9) admits a unique solution on [t k , T ] × R d × A, which satisfies (3.14).

• For k = n-1, we directly get the existence and uniqueness of ϑ π on [t n-1 , T ]×R d ×A from Theorems 3.4 and 3.5 in [START_REF] Barles | BSDEs and integral-partial differential equations[END_REF], and we have

ϑ π = T n-1 π [g] on [t n-1 , T ) × R d × A.
Moreover, we also get by Lemma 3.1:

|ϑ π (t, x, a) -ϑ π (t, x ′ , a ′ )| ≤ max(L 2 , 1) e 2C|π| (1 + |π|) |x -x ′ | + |a -a ′ | for all t ∈ [t n-1 , t n ), (x, a), (x ′ , a ′ ) ∈ R d × A.
• Suppose that the result holds true at step k + 1 i.e. there exists a unique function ϑ π on [t k+1 , T ] × R d × A with linear growth and satisfying (3.8)-(3.9) and (3.14). It remains to prove that ϑ π is uniquely determined by (3.9)

on [t k , t k+1 ) × R d × A and that it satisfies (3.14) on [t k , t k+1 ) × R d × A.
Since ϑ π satisfies (3.14) at time t k+1 , we deduce that the function

ψ k+1 (x) := sup a∈A ϑ π (t k+1 , x, a), x ∈ R d ,
is also Lipschitz continuous, and satisfies by the induction hypothesis:

|ψ k+1 (x) -ψ k+1 (x ′ )| ≤ max(L 2 , 1) e 2C|π| (1 + |π|) n-k-1 |x -x ′ |, (3.15)
for all x, x ′ ∈ R d . Under (H1) and (H2), we can apply Theorems 3.4 and 3.5 in [START_REF] Barles | BSDEs and integral-partial differential equations[END_REF], and we get that ϑ π is the unique viscosity solution with linear growth to (3.9) on [t k , t k+1 )×R d ×A,

with ϑ π = T k π [ψ k+1 ].
Thus it exists and is unique on [t k , T ] × R d × A. From Lemma 3.1 and (3.15), we then get

|ϑ π (t, x, a) -ϑ π (t, x ′ , a ′ )| = |T k π [ψ k+1 ](t, x, a) -T k π [ψ k+1 ](t, x ′ , a ′ )| ≤ max(L 2 , 1) e 2C|π| (1 + |π|) n-k-1 . (1 + |π|)e 2C|π| | |x -x ′ | + |a -a ′ | ≤ max(L 2 , 1) e 2C|π| (1 + |π|) n-k |x -x ′ | + |a -a ′ | for any t ∈ [t k , t k+1 ) and (x, a), (x ′ , a ′ ) ∈ R d × A, which proves the required induction inequality at step k. ✷ Remark 3.1 The function a → ϑ π (t, x, .
) is continuous on A, for each (t, x), and so the function v π is well-defined by (3.10). Moreover, the function ϑ π may be written recursively as:

ϑ π (T, ., .) = g on R d × A, ϑ π = T k π [v π (t k+1 , .)], on [t k , t k+1 ) × R d × A, (3.16
)

for k = 0, . . . , n -1. In particular, ϑ π is continuous on (t k , t k+1 ) × R d × A, k ≤ n -1. ✷
As a consequence of the above proposition, we obtain the uniform Lipschitz property of ϑ π and v π , with a Lipschitz constant independent of π. Corollary 3.1 There exists a constant C (independent of |π|) such that

|ϑ π (t, x, a) -ϑ π (t, x ′ , a ′ )| + |v π (t, x, a) -v π (t, x ′ , a ′ )| ≤ C |x -x ′ | + |a -a ′ | , for all t ∈ [0, T ], x, x ′ ∈ R d , a, a ′ ∈ R d .
Proof. Recalling that n|π| is bounded, we see that the sequence appearing in (3.14):

e 2C|π| (1 + |π|) n-k 0≤k≤n-1
is bounded uniformly in |π| (or n), which shows the required Lipschitz property of ϑ π . Since A is assumed to be compact, this shows in particular that the function v π defined by the relation (3.10) is well-defined and finite. Moreover, by noting that

| sup a∈A ϑ π (t, x, a) -sup a∈A ϑ π (t, x ′ , a)| ≤ sup a∈A |ϑ π (t, x, a) -ϑ π (t, x ′ , a)| for all (t, x) ∈ [0, T ] × R d ,
(Y π , Y π , Z π , U π ) in S 2 × S 2 × L 2 (W ) × L 2 (μ). Moreover we have Y π t = ϑ π (t, X t , I t ), and
Y π t = v π (t, X t , I t ) (3.17) for all t ∈ [0, T ].
Proof. We prove by backward induction on k that (Y π , Y π , Z π , U π ) is well defined and satisfies (3.17) on [t k , T ].

• Suppose that k = n -1. From Corollary 2.3 in [START_REF] Barles | BSDEs and integral-partial differential equations[END_REF], we know that (Y π , Z π , U π ), exists and is unique on [t n-1 , T ]. Moreover, from Theorems 3.4 and 3.5 in [START_REF] Barles | BSDEs and integral-partial differential equations[END_REF], we get

Y π t = T k π [g](t, X t , I t ) = ϑ π (t, X t , I t ) on [t n-1 , T ]
. By (3.7), we then have for all t ∈ [t n-1 , T ):

Y π t = 1 (t n-1 ,T ) (t) ϑ π (t, X t , I t ) + 1 t n-1 (t) ess sup a∈A ϑ π (t, X t , a) = 1 (t n-1 ,T ) (t) ϑ π (t, X t , I t ) + 1 t n-1 (t) sup a∈A ϑ π (t, X t , a) = v π (t, X t , I t ),
since the essential supremum and supremum coincide by continuity of a → ϑ π (t, X t , a) on the compact set A.

• Suppose that the result holds true for some k ≤ n -1. Then, we see that (Y π , Z π , U π ) is defined on [t k-1 , t k ) as the solution to a BSDE driven by W and μ with a terminal condition v π (t k , X t k ). Since v π satisfies a linear growth condition, we know again by Corollary 2.3 in [START_REF] Barles | BSDEs and integral-partial differential equations[END_REF] that (Y π , Z π , U π ), thus also Y π , exists and is unique on [t k-1 , t k ). Moreover, using again Theorems 3.4 and 3.5 in [START_REF] Barles | BSDEs and integral-partial differential equations[END_REF], we get (3.17) on [t k-1 , t k ). ✷

We end this section with a conditional regularity result for the discretely jump-constrained BSDE.

Proposition 3.3 There exists some constant C such that

sup t∈[t k ,t k+1 ) E t k |Y π t -Y π t k | 2 + sup t∈(t k ,t k+1 ] E t k |Y π t -Y π t k+1 | 2 ≤ C(1 + |X t k | 2 )|π|,
for all k = 0, . . . , n -1.

Proof. Fix k ≤ n -1. By Itô's formula, we have for all t ∈ [t k , t k+1 ):

E t k |Y π t -Y π t k | 2 = 2E t k t t k f (X s , I s , Y π s , Z π s )(Y π t k -Y π s )ds + E t k t t k |Z π s | 2 + E t k t t k A |U π s (a)| 2 λ(da)ds ≤ E t k t t k |Y π s -Y π t k | 2 + C|π| 1 + E t k sup s∈[t k ,t k+1 ] |X s | 2 + C|π|E t k sup s∈[t k ,t k+1 ] |Y π s | 2 + |Z π s | 2 + A |U π s (a)| 2 λ(da) ,
by the linear growth condition on f (recall also that A is compact), and Young inequality. Now, by standard estimate for X under growth linear condition on b and σ, we have:

E t k sup s∈[t k ,t k+1 ] |X s | 2 ≤ C(1 + |X t k | 2 ). (3.18)
We also know from Proposition 4.2 in [START_REF] Bouchard | Discrete time approximation of decoupled FBSDE with jumps[END_REF], under (H1) and (H2), that there exists a constant C depending only on the Lipschitz constants of b, σ f and v π (t k+1 , .) (which does not depend on π by Corollary 3.1), such that

E t k sup s∈[t k ,t k+1 ] |Y π s | 2 + |Z π s | 2 + A |U π s (a)| 2 λ(da) ≤ C(1 + |X t k | 2 ). (3.19)
We deduce that

E t k |Y π t -Y π t k | 2 ≤ E t k t t k |Y π s -Y π t k | 2 + C|π|(1 + |X t k | 2 ),
and we conclude for the regularity of Y π by Gronwall's lemma. Finally, from the definition (3.6)-(3.7) of Y π and Y π , Itô isometry for stochastic integrals, and growth linear condition on f , we have for all t ∈ (t k , t k+1 ):

E t k |Y π t -Y π t k+1 | 2 = E t k |Y π t -Y π t k+1 | 2 ≤ 3E t k t k+1 t k |f (X s , I s , Y π s , Z π s )| 2 + |Z π s | 2 + A |U π s (a)| 2 λ(da) ds ≤ C|π|E t k 1 + sup s∈[t k ,t k+1 ] |X s | 2 + |Y π s | 2 + |Z π s | 2 + A |U π s (a)| 2 λ(da) ≤ C|π|(1 + |X t k | 2 ),
where we used again (3.18) and (3.19). This ends the proof. ✷

Convergence of discretely jump-constrained BSDE

This section is devoted to the convergence of the discretely jump-constrained BSDE towards the minimal solution to the BSDE with nonpositive jump.

Convergence result

Lemma 4.1 We have the following assertions:

1) The familly (ϑ π ) π is nondecreasing and upper bounded by v: for any grids π and π ′ such that π ⊂ π ′ , we have

ϑ π (t, x, a) ≤ ϑ π ′ (t, x, a) ≤ v(t, x) , (t, x, a) ∈ [0, T ] × R d × A .
2) The familly (ϑ π ) π satisfies a uniform linear growth condition: there exists a constant C such that

|ϑ π (t, x, a)| ≤ C(1 + |x|),
for any (t, x, a) ∈ [0, T ] × R d × A and any grid π.

Proof. 1) Let us first prove that ϑ π ≤ v. Since v is a (continuous) viscosity solution to the HJB equation (3.3), and v does not depend on a, we see that v is a viscosity supersolution to the IPDE in (3.9) satisfied by ϑ π on each interval [t k , t k+1 ). Now, since v(T, x) = ϑ π (T, x, a), we deduce by comparison principle for this IPDE (see e.g. Theorem 3.4 in [START_REF] Barles | BSDEs and integral-partial differential equations[END_REF])

on [t n-1 , T ) × R d × A that v(t, x) ≥ ϑ π (t, x, a) for all t ∈ [t n-1 , T ], (x, a) ∈ R d × A. In particular, v(t - n-1 , x) = v(t n-1 , x) ≥ sup a∈A ϑ π (t n-1 , x, a) = ϑ π (t - n-1 , x, a)
. Again, by comparison principle for the IPDE (3.9) on [t n-2 , t n-1 ) × R d × A, it follows that v(t, x) ≥ ϑ π (t, x, a) for all t ∈ [t n-2 , t n-1 ], (x, a) ∈ R d × A. By backward induction on time, we conclude that v ≥ ϑ π on [0, T ] × R d × A.

Let us next consider two partitions π = (t k ) 0≤k≤n and π ′ = (t ′ k ) 0≤k≤n ′ of [0, T ] with π ⊂ π ′ , and denote by m = max{k ≤ n ′ : t ′ m / ∈ π}. Thus, all the points of the grid π and π ′ coincide after time t ′ m , and since ϑ π and ϑ π ′ are viscosity solution to the same IPDE (3.9) starting from the same terminal data g, we deduce by uniqueness that

ϑ π = ϑ π ′ on [t ′ m , T ] × R d × A. Then, we have ϑ π ′ (t ′ - m , x, a) = sup a∈A ϑ π (t ′ m , x, a) = sup a∈A ϑ π (t ′ m , x, a) ≥ ϑ π (t - m , x, a
) since ϑ π is continuous outside of the points of the grid π (recall Remark 3.1). Now, since ϑ π and ϑ π ′ are viscosity solution to the same IPDE (3.9) on [t ′ m-1 , t m ), we deduce by comparison principle that

ϑ π ′ ≥ ϑ π on [t ′ m-1 , t ′ m ] × R d × A. Proceeding by backward induction, we conclude that ϑ π ′ ≥ ϑ π on [0, T ] × R d × A.
2) Denote by π 0 = {t 0 = 0, t 1 = T } the trivial grid of [0, T ]. Since ϑ π 0 ≤ ϑ π ≤ v and ϑ π 0 and v satisfy a linear growth condition, we get (recall that A is compact):

|ϑ π (t, x, a)| ≤ |ϑ π 0 (t, x, a)| + |v(t, x)| ≤ C(1 + |x|),
for any (t, x, a) ∈ [0, T ] × R d × A and any grid π. ✷

In the sequel, we denote by ϑ the increasing limit of the sequence (ϑ π ) π when the grid increases by becoming finer, i.e. its modulus |π| goes to zero. The next result shows that ϑ does not depend on the variable a in A.

Proposition 4.1 The function ϑ is l.s.c. and does not depend on the variable a ∈ A:

ϑ(t, x, a) = ϑ(t, x, a ′ ) , (t, x) ∈ [0, T ] × R d , a, a ′ ∈ A .
To prove this result we use the following lemma. Observe by definition (3.10) of v π that the function v π does not depend on a on the grid times π, and we shall denote by misuse of notation: v π (t k , x), for k ≤ n, x ∈ R d . Lemma 4.2 There exists a constant C (not depending on π) such that Since |π p | → 0 as p → ∞ we get t p → t as p → +∞. We then have from the previous lemma:

|ϑ π (t, x, a) -v π (t k+1 , x)| ≤ C(1 + |x|)|π| 1 2 for all k = 0, . . . , n -1, t ∈ [t k , t k+1 ), (x, a) ∈ R d × A. Proof. Fix k = 0, . . . , n-1, t ∈ [t k , t k+1 ) and (x, a) ∈ R d ×A. Let ( Ỹ, Z, Ũ ) be the solution to the BSDE Ỹs = v π (t k+1 , X t,x,a t k+1 ) + t k+1 s f (X t,
|ϑ π p (t, x, a) -ϑ π p (t, x, a ′ )| ≤ |ϑ π p (t, x, a) -v π p (t p , x)| + |v π p (t p , x) -ϑ π p (t, x, a ′ )| ≤ 2C|π p | 1 2 .
Sending p to ∞ we obtain that ϑ(t, x, a) = ϑ(t, x, a ′ ). ✷ Corollary 4.1 We have the identification: ϑ = v, and the sequence (v π ) π also converges to v.

Proof. We proceed in two steps.

Step 1. The function ϑ is a supersolution to (3.3). Since ϑ π k (T, .) = g for all k ≥ 1, we first notice that ϑ(T, .) = g. Next, since ϑ does not depend on the variable a, we have ϑ π (t, x, a) ↑ ϑ(t, x) as |π| ↓ 0 for any (t, x, a) ∈ [0, T ] × R d × A. Moreover, since the function ϑ is l.s.c, we have

ϑ = ϑ * = lim inf |π|→0 * ϑ π , (4.3)
where lim inf

|π|→0 * ϑ π (t, x, a) := lim inf |π| → 0 (t ′ , x ′ , a ′ ) → (t, x, a) t ′ < T ϑ π (t ′ , x ′ , a ′ ), (t, x, a) ∈ [0, T ] × R d × R q .
Fix now some (t, x) ∈ [0, T ] × R d and a ∈ A and (p, q, M ) ∈ J2,ϑ(t, x), the limiting parabolic subjet of ϑ at (t, x) (see definition in [START_REF] Crandall | User's guide to viscosity solutions of second order partial differential equations[END_REF]). From standard stability results, there exists a sequence (π

k , t k , x k , a k , p k , q k , M k ) k such that (p k , q k , M k ) ∈ J2,-ϑ π k (t k , x k , a k )
for all k ≥ 1 and

(t k , x k , a k , ϑ π k (t k , x k , a k )) -→ (t, x, a, ϑ(t, x, a)) as k → ∞, |π k | → 0.
From the viscosity supersolution property of ϑ π k to (3.9) in terms of subjets, we have

-p k -b(x k , a k ).q k - 1 2 tr(σσ ⊺ (x k , a k )M k ) -f x k , a k , ϑ π k (t k , x k , a k ), σ ⊺ (x k , a k )q k ) - A ϑ π k (t k , x k , a ′ ) -ϑ π k (t k , x k , a k ) λ(da ′ ) ≥ 0
for all k ≥ 1. Sending k to infinity and using (4.3), we get -p -b(x, a).q -1 2 tr(σσ ⊺ (x, a)M ) -f x, a, ϑ(t, x), σ ⊺ (x, a)q) ≥ 0.

Since a is arbitrary in A, this shows

-p -sup a∈A b(x, a).q + 1 2 tr(σσ ⊺ (x, a)M ) + f x, a, ϑ(t, x), σ ⊺ (x, a)q) ≥ 0,
i.e. the viscosity supersolution property of ϑ to (3.3).

Step 2. Comparison. Since the PDE (3.3) satisfies a comparison principle, we have from the previous step ϑ ≥ v, and we conclude with Lemma 4.1 that ϑ = v. Finally, by definition (3.10) of v π and from Lemma 4.1, we clearly have ϑ π ≤ v π ≤ v, which also proves that (v π ) π converges to v. ✷

In terms of the discretely jump-constrained BSDE, the convergence result is formulated as follows:

Proposition 4.2 We have Y π t ≤ Y π t ≤ Y t , 0 ≤ t ≤ T , and
E sup t∈[0,T ] |Y t -Y π t | 2 + E sup t∈[0,T ] |Y t -Y π t | 2 + E T 0 |Z t -Z π t | 2 dt → 0,
as |π| goes to zero.

Proof. Recall from (3.2) and (3.17) that we have the representation:

Y t = v(t, X t ), Y π t = v π (t, X t , I t ), Y π t = ϑ(t, X t , I t ), (4.4)
and the first assertion of Lemma (4.1), we clearly have:

Y π t ≤ Y π t ≤ Y t , 0 ≤ t ≤ T .
The convergence in S 2 for Y π to Y and Y π to Y comes from the above representation (4.4), the pointwise convergence of ϑ π and v π to v in Corollary 4.1, and the dominated convergence theorem by recalling that 0

≤ (v -v π )(t, x, a) ≤ (v -ϑ π )(t, x, a) ≤ v(t, x) ≤ C(1 + |x|).
Y π t = g(X T ) + T t f (X s , I s , Y π s , Z π s ) - T t Z π s dW s - T t A U π s (a)μ(ds, da) + K π T -K π t ,
where K π is the nondecreasing process defined by:

K π t = t k ≤t (Y π t k -Y π t k ), for t ∈ [0, T ]. Denote by δY = Y -Y π , δZ = Z -Z π , δU = U -U π and δK = K -K π . From Itô's
formula, Young Inequality and (H2), there exists a constant C such that

E |δY t | 2 + 1 2 E T t |δZ s | 2 ds + 1 2 E T t |δU s (a)| 2 λ(da)ds ≤ C T t E |δY s | 2 ds + 1 ε E sup s∈[0,T ] |δY s | 2 + εE δK T -δK t 2 (4.5)
for all t ∈ [0, T ], with ε a constant to be chosen later. From the definition of δK we have

δK T -δK t = δY t - T t f (X s , I s , Y s , Z s ) -f (X s , I s , Y π s , Z π s ) ds + T 0 δZ s dW s + T t A
δU s (a)μ(ds, da) .

Therefore, by (H2), we get the existence of a constant C ′ such that

E δK T -δK t 2 ≤ C ′ E sup s∈[0,T ] |δY s | 2 + E T t |δZ s | 2 ds + E T t |δU s (a)| 2 λ(da)ds
Taking ε = C ′ 4 and plugging this last inequality in (4.5), we get the existence of a constant C ′′ such that

E T t |δZ s | 2 ds + E T t |δU s (a)| 2 λ(da)ds ≤ C ′′ E sup s∈[0,T ] |δY s | 2 , (4.6)
which shows the L 2 (W ) convergence of Z π to Z from the S 2 convergence of Y π to Y . ✷

Rate of convergence

We next provide an error estimate for the convergence of the discretely jump-constrained BSDE. We shall combine BSDE methods and PDE arguments adapted from the shaking coefficients approach of Krylov [START_REF] Krylov | On the rate of convergence of finite difference approximations for Bellman's equations with variable coefficients[END_REF] and switching systems approximation of Barles, Jacobsen [START_REF] Barles | Error bounds for monotone approximation schemes for parabolic Hamilton-Jacobi-Bellman equations[END_REF]. We make further assumptions: (H2') The function f does not depend on z: f (x, a, y, z) = f (x, a, y) for all (x, a, y, z

) ∈ R d × A × R × R d and
(i) the functions f (., ., 0) and g are uniformly bounded:

sup x∈R d ,a∈A |f (x, a, 0)| + |g(x)| < ∞, (ii) for all (x, a) ∈ R d × A, y → f (x, a, y) is convex.
Under these assumptions, we obtain the rate of convergence for v π and ϑ π towards v.

Theorem 4.1 Under (H1') and (H2'), there exists a constant C such that

0 ≤ v(t, x) -v π (t, x, a) ≤ v(t, x) -ϑ π (t, x, a) ≤ C|π| 1 10
for all (t, x, a) ∈ [0, T ] × R d × A and all grid π with |π| ≤ 1. Moreover, when f (x, a) does not depend on y, the rate of convergence is improved to |π| 1 6 .

Before proving this result, we give as corollary the rate of convergence for the discretely jump-constrained BSDE.

Corollary 4.2 Under (H1') and (H2'), there exists a constant C such that

E sup t∈[0,T ] |Y t -Y π t | 2 + E sup t∈[0,T ] |Y t -Y π t | 2 + E T 0 |Z t -Z π t | 2 dt ≤ C|π| 1 5
.

for all grid π with |π| ≤ 1, and the above rate is improved to |π| Proof. From the representation (4.4), and Theorem 4.1, we immediately have

E sup t∈[0,T ] |Y t -Y π t | 2 + E sup t∈[0,T ] |Y t -Y π t | 2 ≤ C|π| 1 5 , (4.7)
(resp. |π| is the optimal rate that one can prove in our generalized stochastic control context with fully nonlinear HJB equation by PDE approach and shaking coefficients technique, see [START_REF] Krylov | On the rate of convergence of finite difference approximations for Bellman's equations with variable coefficients[END_REF], [START_REF] Barles | Error bounds for monotone approximation schemes for parabolic Hamilton-Jacobi-Bellman equations[END_REF], [START_REF] Fahim | A Probabilistic Numerical Scheme for Fully Nonlinear PDEs[END_REF] or [START_REF] Tan | A splitting method for fully nonlinear degenerate parabolic PDEs[END_REF]. However, this rate may not be the sharpest one. In the case of continuously reflected BSDEs, i.e. BSDEs with upper or lower constraint on Y , it is known that Y can be approximated by discretely reflected BSDEs, i.e. BSDEs where reflection on Y operates a finite set of times on the grid π, with a rate |π| 1 2 (see [START_REF] Bally | Error analysis of the quantization algorithm for obstacle problems[END_REF]). The standard arguments for proving this rate is based on the representation of the continuously (resp. discretely) reflected BSDE as optimal stopping problems where stopping is possible over the whole interval time (resp. only on the grid times). In our jump-constrained case, we know from [START_REF] Kharroubi | Feynman-Kac representation for Hamilton-Jacobi-Bellman IPDE[END_REF] that the minimal solution to the BSDE with nonpositive jumps has the stochastic control representation (1.2) when f (x, a) does not depend on y and z. We shall prove an analog representation for discretely jumpconstrained BSDEs, and this helps to improve the rate of convergence from |π| 1 10 to |π| 1 6 . ✷ The rest of this section is devoted to the proof of Theorem 4.1. We first consider the special case where f (x, a) does not depend on y, and then address the case f (x, a, y).

Proof of Theorem 4.1 in the case f (x, a). In the case where f (x, a) does not depend on y, z, by (linear) Feynman-Kac formula for ϑ π solution to (3.9), and by definition of v π in (3.10), we have:

v π (t k , x) = sup a∈A E t k+1 t k f (X t k ,x,a t , I t k ,a t )dt + v π (t k+1 , X t k ,x,a t k+1 ) , k ≤ n -1, x ∈ R d .
By induction, this dynamic programming relation leads to the following stochastic control problem with discrete time policies:

v π (t k , x) = sup α∈A π F E T t k f ( Xt k ,x,α t , Īα t )dt + g( Xt k ,x,α T ) ,
where A π F is the set of discrete time processes α = (α t j ) j≤n-1 , with α t j F t j -measurable, valued in A, and

Xt k ,x,α t = x + t t k b( Xt,x,α s , Īα s )ds + t t k σ( Xt k ,x,α s , Īα s )dW s , t k ≤ t ≤ T, Īα t = α t j + (t j ,t] A (a -Īα s -)µ(ds, da), t j ≤ t < t j+1 , j ≤ n -1.
In other words, v π (t k , x) corresponds to the value function for a stochastic control problem where the controller can act only at the dates t j of the grid π, and then let the regime of the coefficients of the diffusion evolve according to the Poisson random measure µ. Let us introduce the following stochastic control problem with piece-wise constant control policies:

ṽπ (t k , x) = sup α∈A π F E T t k f ( Xt k ,x,α t , Ĩα t )dt + g( Xt k ,x,α T ) ,
where for α = (α

t j ) j≤n-1 ∈ A π F : Xt k ,x,α t = x + t t k b( Xt,x,α s , Ĩα s )ds + t t k σ( Xt k ,x,α s , Ĩα s )dW s , t k ≤ t ≤ T, Ĩα t = α t j , t j ≤ t < t j+1 , j ≤ n -1.
It is shown in [START_REF] Krylov | Approximating value functions for controlled degenerate diffusion processes by using piece-wise constant policies[END_REF] that ṽπ approximates the value function v for the controlled diffusion problem (1.2), solution to the HJB equation (3.3), with a rate |π|

1 6 : 0 ≤ v(t k , x) -ṽπ (t k , x) ≤ C|π| 1 6 , (4.8)
for all t k ∈ π, x ∈ R d . Now, recalling that A is compact and λ(A) < ∞, it is clear that there exists some positive constant C such that for all α ∈ A π F , j ≤ n -1:

E sup t∈[t j ,t j+1 ) | Īα t -Ĩα t | 2 ≤ C|π|,
and then by standard arguments under Lipschitz condition on b, σ:

E sup t∈[t j ,t j+1 ] | Xt k ,x,α t -Xt k ,x,α t | 2 ≤ C|π|, k ≤ j ≤ n -1, x ∈ R d .
By the Lipschitz conditions on f and g, it follows that

|v π (t k , x) -ṽπ (t k , x)| ≤ C|π| 1 2 ,
and thus with (4.8):

0 ≤ sup x∈R d (v -v π )(t k , x) ≤ C|π| 1 6 .

21

Finally, by combining with the estimate in Lemma 4.2, which gives actually under (H2')(i):

|ϑ π (t, x, a) -v π (t k+1 , x)| ≤ C|π| 1 2 , t ∈ [t k , t k+1 ), (x, a) ∈ R d × A,
together with the 1/2-Hölder property of v in time (see (3.4)), we obtain:

sup (t,x,a)∈[0,T ]×R d ×A (v -ϑ π )(t, x, a) ≤ C(|π| 1 6 + |π| 1 2 ) ≤ C|π| 1 6 ,
for |π| ≤ 1. This ends the proof. ✷

Let us now turn to the case where f (x, a, y) may also depend on y. We cannot rely anymore on the convergence rate result in [START_REF] Krylov | Approximating value functions for controlled degenerate diffusion processes by using piece-wise constant policies[END_REF]. Instead, recalling that A is compact and since σ, b and f are Lipschitz in (x, a), we shall apply the switching system method of Barles and Jacobsen [START_REF] Barles | Error bounds for monotone approximation schemes for parabolic Hamilton-Jacobi-Bellman equations[END_REF], which is a variation of the shaken coefficients method and smoothing technique used in Krylov [START_REF] Krylov | On the rate of convergence of finite difference approximations for Bellman's equations with variable coefficients[END_REF], in order to obtain approximate smooth subsolution to (3.3). By Lemmas 3.3 and 3.4 in [START_REF] Barles | Error bounds for monotone approximation schemes for parabolic Hamilton-Jacobi-Bellman equations[END_REF], one can find a family of smooth functions (w

ε ) 0<ε≤1 on [0, T ] × R d such that: sup [0,T ]×R d |w ε | ≤ C, (4.9)
sup

[0,T ]×R d |w ε -w| ≤ Cε 1 3 , (4.10) sup
[0,T ]×R d |∂ β 0 t D β w ε | ≤ Cε 1-2β 0 -d i=1 β i , β 0 ∈ N, β = (β 1 , . . . , β d ) ∈ N d , (4.11)
for some positive constant C independent of ε, and by convexity of f in (H2')(ii), for any ε ∈ (0, 1], (t, x) ∈ [0, T ] × R d , there exists a t,x,ε ∈ A such that:

-L at,x,ε w ε (t, x) -f (x, a t,x,ε , w ε (t, x)) ≥ 0. (4.12)
Recalling the definition of the operator T k π in (3.11), we define for any function ϕ on [0, T ] × R d × A, Lipschitz in (x, a):

T π [ϕ](t, x, a) := T k π [ϕ(t k+1 , ., .)](t, x, a), t ∈ [t k , t k+1 ), (x, a) ∈ R d × A,
for k = 0, . . . , n -1, and

S π [ϕ](t, x, a) := 1 |π| ϕ(t, x) -T π [ϕ](t, x, a) +(t k+1 -t) L a ϕ(t, x) + f (x, a, ϕ(t, x) , for (t, x, a) ∈ [t k , t k+1 ) × R d × A, k ≤ n -1.
We have the following key error bound on S π .

Lemma 4.3 Let (H1') and (H2')(i) hold. There exists a constant C such that

|S π [ϕ ε ](t, x, a)| ≤ C |π| 1 2 (1 + ε -1 ) + |π|ε -3 , (t, x, a) ∈ [0, T ] × R d × A,
for any family (ϕ ε ) ε of smooth functions on [0, T ] × R d satisfying (4.9) and (4.11).

Proof.

Fix (t, x, a) ∈ [0, T ] × R d × A. If t = T , we have |S π [ϕ ε ](t, x, a)| = 0. Suppose that t < T and fix k ≤ n such that t ∈ [t k , t k+1
). Given a smooth function ϕ ε satisfying (4.9) and (4.11), we split:

|S π [ϕ ε ](t, x, a)| ≤ A ε (t, x, a) + B ε (t, x, a),
where

A ε (t, x, a) := 1 |π| T π [ϕ ε ](t, x, a) -E ϕ ε (t k+1 , X t,x,a t k+1 ) -(t k+1 -t)f (x, a, ϕ ε (t, x) ,
and

B ε (t, x, a) := 1 |π| E ϕ ε (t k+1 , X t,x,a t k+1 ) -ϕ ε (t, x) -(t k+1 -t)L a ϕ ε (t, x) ,
and we study each term A ε and B ε separately.

1. Estimate on A ε (t, x, a). Define (Y ϕε , Z ϕε , U ϕε ) as the solution to the BSDE on [t, t k+1 ]:

Y ϕε s = ϕ ε (t k+1 , X t,x,a t k+1 ) + t k+1 s f (X t,x,a r , I t,a r , Y ϕε r )dr - t k+1 s Z ϕε r dW r - t k+1 s A U ϕε r (a)μ(dr, da) , s ∈ [t, t k+1 ]. (4.13)
From Theorems 3.4 and 3.5 in [START_REF] Barles | BSDEs and integral-partial differential equations[END_REF], we have

Y ϕε t = T π [ϕ ε ](t,
x, a), and by taking expectation in (4.13), we thus get:

Y ϕε t = T π [ϕ ε ](t, x, a) = E ϕ ε (t k+1 , X
|Z ϕε s | 2 ] + E sup s∈[t,t k+1 ] A |U ϕε s (a)| 2 λ(da)] ≤ C.
From (4.11), we then have:

E sup s∈[t,t k+1 ] |Y ϕε s -ϕ ε (t, x)| ≤ C |t k+1 -t|ε -1 + E |X t,x,a t k+1 -x| + |π| ≤ C|π| 1 2 1 + ε -1 ,
by (4.14). This leads to the error bound for A ε (t, x, a):

A ε (t, x, a) ≤ C|π| 1 2 (1 + ε -1 ).
2. Estimate on B ε (t, x, a). From Itô's formula we have

B ε (t, x, a) = 1 |π| E t k+1 t L I t,a s ϕ ε (s, X t,x,a s ) -L a ϕ ε (t, x) ds ≤ B 1 ε (t, x, a) + B 2 ε (t, x, a)
where

B 1 ε (t, x, a) = 1 |π| E t k+1 t b(X t,x,a s , I t,a s -b(x, a)).D x ϕ ε (s, X t,x,a s ) + 1 2 tr σσ ⊺ (X t,x,a s , I t,a s ) -σσ ⊺ (x, a) D 2 x ϕ ε (t, x) ds and B 2 ε (t, x, a) = 1 |π| E t k+1 t La t,x ϕ ε (s, X t,x,a s ) -La t,x ϕ ε (t, x) ds , with La t,x defined by La t,x ϕ ε (t ′ , x ′ ) = ∂ϕ ε ∂t (t ′ , x ′ ) + b(x, a).D x ϕ ε (t ′ , x ′ ) + 1 2 tr σσ ⊺ (x, a)D 2 x ϕ ε (t ′ , x ′ ) .
Under (H1), (H1'), and by (4.11), we have

B 1 ε (t, x, a) ≤ C(1 + ε -1 )E sup s∈[t,t k+1 ] |X t,x,a s -x| + |I t,a s -a| ≤ C(1 + ε -1 )|π| 1 2 ,
where we used again (4.14). On the other hand, since ϕ ε is smooth, we have from Itô's formula

B 2 ε (t, x, a) = 1 |π| E t k+1 t s t L I t,a r La t,x φ(r, X t,x,a r
)dr ds .

Under (H1') and by (4.11), we then see that

B 2 ε (t, x, a) ≤ C|π|ε -3 ,
and so:

B ε (t, x, a) ≤ C |π| 1 2 (1 + ε -1 ) + |π|ε -3 .
Together with the estimate for A ε (t, x, a), this proves the error bound for |S π [ϕ ε ](t, x, a)|. ✷

We next state a maximum principle type result for the operator T π .

Lemma 4.4 Let ϕ and ψ be two functions on [0, T ] × R d × A, Lipschitz in (x, a). Then, there exists some positive constant C independent of π such that sup

(t,x,a)∈[t k ,t k+1 ]×R d ×A (T π [ϕ] -T π [ψ])(t, x, a) ≤ e C|π| sup (x,a)∈R d ×A (ϕ -ψ)(t k+1 , x, a) ,
for all k = 0, . . . , n -1.

Proof. Fix k ≤ n -1, and set

M := sup (x,a)∈R d ×A (ϕ -ψ)(t k+1 , x, a).
We can assume w.l.o.g. that M < ∞ since otherwise the required inequality is trivial. Let us denote by ∆v the function

∆v(t, x, a) = T π [ϕ](t, x, a) -T π [ψ](t, x, a), for all (t, x, a) ∈ [t k , t k+1 ] × R d × A.
By definition of T π , and from the Lipschitz condition of f , we see that ∆v is a viscosity subsolution to

     -L a ∆v(t, x, a) -C |∆v(t, x, a)| + |D∆v(t, x, a)| -A ∆v(t, x, a ′ ) -∆v(t, x, a) λ(da ′ ) = 0, for (t, x, a) ∈ [t k , t k+1 ) × R d × A, ∆v(t k+1 , x, a) ≤ M , for (x, a) ∈ R d × A . ( 4
.15) Then, we easily check that the function Φ defined by

Φ(t, x, a) = M e C(t k+1 -t) , (t, x, a) ∈ [t k , t k+1 ] × R d × A , is a solution to      -L a Φ(t, x, a) -C |Φ(t, x, a)| + |DΦ(t, x, a)| -A Φ(t, x, a ′ ) -Φ(t, x, a) λ(da ′ ) = 0, for (t, x, a) ∈ [t k , t k+1 ) × R d × A, Φ(t k+1 , x, a) = M , for (x, a) ∈ R d × A . (4.16)
From the comparison theorem in [START_REF] Barles | BSDEs and integral-partial differential equations[END_REF] for viscosity solutions of semi-linear IPDEs, we get that ∆v ≤ Φ on [t k , t k+1 ] × R d × A, which proves the required inequality. ✷

Proof of Theorem 4.1. By (3.10) and (3.16), we observe that v π is a fixed point of T π , i.e.

T π [v π ] = v π .
On the other hand, by (4.12), and the estimate of Lemma 4.3 applied to w ε , we have:

w ε (t, x) -T π [w ε ](t, x, a t,x,ε ) ≤ |π|S π [w ε ](t, x, a t,x,ε ) ≤ C|π| S(π, ε)
where we set: S(π, ε) = (|π|

3 2 (1 + ε -1 ) + |π| 2 ε -3 ). Fix k ≤ n -1. By Lemma 4.4, we then have for all t ∈ [t k , t k+1 ], x ∈ R d : w ε (t, x) -v π (t, x, a t,x,ε ) = w ε (t, x) -T π [w ε ](t, x, a t,x,ε ) + (T π [w ε ] -T π [v π ])(t, x, a t,x,ε ) ≤ C|π| S(π, ε) + e C|π| sup (x,a)∈R d ×A (w ε -v π )(t k+1 , x, a). (4.17)
Recalling by its very definition that v π does not depend on a ∈ A on the grid times of π, and denoting then M k := sup x∈R d (w ε -v π )(t k , x), we have by (4.17) the relation:

M k ≤ C|π| S(π, ε) + e C|π| M k+1 .
By induction, this yields:

sup x∈R d (w ε -v π )(t k , x) ≤ C e Cn|π| -1 e C|π| -1 |π| S(π, ε) + e Cn|π| sup x∈R d (w ε -v π )(T, x) ≤ C S(π, ε) + C sup x∈R d (w ε -v)(T, x),
since n|π| is bounded and v(T, x) = v π (T, x) (= g(x)). From (4.10), we then get:

sup x∈R d (v -v π )(t k , x) ≤ C ε 1 3 + |π| 1 2 (1 + ε -1 ) + |π|ε -3 .
By minimizing the r.h.s of this estimate with respect to ε, this leads to the error bound when taking ε = |π|

3 10 ≤ 1: sup x∈R d (v -v π )(t k , x) ≤ C|π| 1 10
.

Finally, by combining with the estimate in Lemma 4.2, which gives actually under (H2')(i):

|ϑ π (t, x, a) -v π (t k+1 , x)| ≤ C|π| 1 2 , t ∈ [t k , t k+1 ), (x, a) ∈ R d × A,
together with the 1/2-Hölder property of v in time (see (3.4)), we obtain:

sup (t,x,a)∈[0,T ]×R d ×A (v -ϑ π )(t, x, a) ≤ C(|π| 1 10 + |π| 1 2 ) ≤ C|π| 1 10 .
This ends the proof. ✷

Approximation scheme for jump-constrained BSDE and stochastic control problem

We consider the discrete time approximation of the discretely jump-constrained BSDE in the case where f (x, a, y) does not depend on z, and define the scheme ( Ȳ π , Ȳπ , Zπ ) by induction on the grid π = {t 0 = 0 < . . . < t k < . . . < t n = T } by:

       Ȳ π T = Ȳπ T = g( Xπ T ) Ȳπ t k = E t k Ȳ π t k+1 + f ( Xπ t k , I t k , Ȳπ t k )∆t k Ȳ π t k = ess sup a∈A E t k ,a Ȳπ t k , k = 0, . . . , n -1, ( 5
E t k Ȳ π t k+1 = E Ȳ π t k+1 Xπ t k , I t k and E t k ,a Ȳπ t k ] = E Ȳπ t k Xπ t k , I t k = a .
We then have:

Ȳπ t k = θπ k ( Xπ t k , I t k ), Y π t k = vπ k ( Xπ t k ),
for some sequence of functions ( θπ k ) k and (v π k ) k defined respectively on R d × A and R d by backward induction:

     vπ n (x, a) = θπ n (x) = g(x) θπ k (x, a) = E vπ k+1 ( Xπ t k+1 , I t k+1 ) ( Xπ t k , I t k ) = (x, a) + f (x, a, θπ k (x, a))∆t k vπ k (x) = sup a∈A θπ k (x, a), k = 0, . . . , n -1.
(5.2) There are well-known different methods (Longstaff-Schwartz least square regression, quantization, Malliavin integration by parts, see e.g. [START_REF] Bally | Error analysis of the quantization algorithm for obstacle problems[END_REF], [START_REF] Gobet | Rate of convergence of empirical regression method for solving generalized BSDE[END_REF], [START_REF] Bouchard | Discrete time approximation and Monte-Carlo simulation of backward stochastic differential equations[END_REF]) for computing the above conditional expectations, and so the functions θπ k and vπ k . It appears that in our context, the simulation-regression method on basis functions defined on R d × A, is quite suitable since it allows us to derive at each time step k ≤ n -1, a functional form âk (x), which attains the supremum over A in θπ k (x, a). We shall see later in this section that the feedback control (â k (x)) k provides an approximation of the optimal control for the HJB equation associated to a stochastic control problem when f (x, a) does not depend on y. We refer to our companion paper [START_REF] Kharroubi | Numerical algorithm for fully nonlinear HJB equations: an approach by control randomization[END_REF] for the details about the computation of functions θπ k , vπ k , âk by simulation-regression methods, and the associated error analysis.

Error estimate for the discrete time scheme

The main result of this section is to state an error bound between the component Y π of the discretely jump-constrained BSDE and the solution ( Ȳ π , Ȳπ ) to the above discrete time scheme.

Theorem 5.1 There exists some constant C such that:

E Y π t k -Ȳ π t k 2 + sup t∈(t k ,t k+1 ] E Y π t -Ȳ π t k+1 2 + sup t∈[t k ,t k+1 ) E Y π t -Ȳπ t k 2 ≤ C|π|,
for all k = 0, . . . , n -1.

The above convergence rate |π| 1 2 in the L 2 -norm for the discretization of the discretely jump-constrained BSDE is the same as for standard BSDE, see [START_REF] Bouchard | Discrete time approximation and Monte-Carlo simulation of backward stochastic differential equations[END_REF], [START_REF] Zhang | A numerical scheme for BSDEs[END_REF]. By combining with the convergence result in Section 4, we finally obtain an estimate on the error due to the discrete time approximation of the minimal solution Y to the BSDE with nonpositive jumps. We split the error between the positive and negative parts:

Err π + (Y ) := max k≤n-1 E Y t k -Ȳ π t k 2 + + sup t∈(t k ,t k+1 ] E Y t -Ȳ π t k+1 2 + + sup t∈[t k ,t k+1 ) E Y t -Ȳπ t k 2 + 1 2 Err π -(Y ) := max k≤n-1 E Y t k -Ȳ π t k 2 -+ sup t∈(t k ,t k+1 ] E Y t -Ȳ π t k+1 2 -+ sup t∈[t k ,t k+1 ) E Y t -Ȳπ t k 2 - 1 2 .
Corollary 5.1 We have:

Err π -(Y ) ≤ C|π| 1 2 .
Moreover, under (H1') and (H2'),

Err π + (Y ) ≤ C|π| 1 10 ,
and when f (x, a) does not depend on y: In the particular case where f depends only on (x, a), our discrete time approximation scheme is a probabilistic scheme for the fully nonlinear HJB equation associated to the stochastic control problem (1.2). As in [START_REF] Krylov | On the rate of convergence of finite difference approximations for Bellman's equations with variable coefficients[END_REF], [START_REF] Barles | Error bounds for monotone approximation schemes for parabolic Hamilton-Jacobi-Bellman equations[END_REF] or [START_REF] Fahim | A Probabilistic Numerical Scheme for Fully Nonlinear PDEs[END_REF], we have non symmetric bounds on the rate of convergence. For instance, in [START_REF] Fahim | A Probabilistic Numerical Scheme for Fully Nonlinear PDEs[END_REF], the authors obtained a convergence rate |π| ✷ and notice that Xπ = X π, α, where α ∈ A π is a feedback control defined by: αt k = âk ( Xπ t k ) = âk (X π, α t k ), k = 0, . . . , n.

Err π + (Y ) ≤ C|π| 1 6 . Proof. Recall from Proposition 4.2 that Y π t ≤ Y π t ≤ Y t , 0 ≤ t ≤ T . Then, we have: (Y t k -Ȳ π t k ) -≤ |Y π t k -Ȳ π t k |, (Y t -Ȳ π t k+1 ) -≤ |Y π t -Ȳ π t k+1 |, and (Y t k -Ȳπ t k ) -≤ |Y π t k -Ȳπ t k |, for all k ≤ n -1,
Next, we observe that the conditional law of Xπ t k+1 given ( Xπ t k = x, I t k = âk ( Xπ t k ) = âk (x)) is the same than the conditional law of X π, α t k+1 given X π, α t k = x, for k ≤ n -1, and thus the induction step in the scheme (5.1) or (5.2) reads as:

vπ k (X π, α t k ) = E vπ k+1 (X π, α t k+1 ) X π, α t k + f (X π, α t k , αt k )∆t k , k ≤ n -1.
By induction, and law of iterated conditional expectations, we then get: Ȳ π 0 = vπ 0 (X 0 ) = J π (α).

(5.11)

Consider the continuous time piecewise-constant interpolation α ∈ A defined by: αt = αt k , for t ∈ [t k , t k+1 ). By (5.9), (5.10), (5.11), and Corollary 5.1, we finally obtain:

0 ≤ V 0 -J(α) = Y 0 -Ȳ π 0 + J π (α) -J(α) ≤ C|π| 1 6 + C|π| 1 2 ≤ C|π| 1 6 ,
for |π| ≤ 1. In other words, for any small ε > 0, we obtain an ε-approximate optimal control α for the stochastic control problem (5.9) by taking |π| of order ε 6 .

1 4 on one side and |π| 1 10

 11 on the other side under a nondegeneracy condition.

  ) for all t ∈ [t k , t k+1 ) and k = 0 . . . , n -1. The rest of this section is devoted to the proof of existence and uniqueness of a solution to (3.8)-(3.9)-(3.10), together with some uniform Lipschitz properties, and its connection to the discretely jump-constrained BSDE (3.5)-(3.6)-(3.7).

  we also obtain the required Lipschitz property for v π . ✷We now turn to the existence of a solution to the discretely jump-constrained BSDE. Proposition 3.2 The BSDE (3.5)-(3.6)-(3.7) admits a unique solution

  Let us now turn to the component Z. By definition (3.5)-(3.6)-(3.7) of the discretely jump-constrained BSDE we notice that Y π can be written on [0, T ] as:

(

  H1') The functions b and σ are uniformly bounded: sup x∈R d ,a∈A |b(x, a)| + |σ(x, a)| < ∞.

1 3

 1 when f (x, a) does not depend on y.

1 3Remark 4 . 1

 141 when f (x, a) does not depend on y). Finally, the convergence rate for Z follows from the inequality (4.6).✷ The above convergence rate |π| 1 10

  and t ∈ [0, T ]. The error bound on Err π -(Y ) follows then from the estimation in Theorem 5.1. The error bound on Err π -(Y ) follows from Corollary 4.2 and Theorem 5.1. ✷ Remark 5.1

1 4 on one side and |π| 1 10 2 for one side, and |π| 1 6 1 6 1 .

 112111 on the other side, while we improve the rate to |π| 1 on the other side. This induces a global error Err π (Y ) = Err π + (Y ) + Err π -(Y ) of order |π| , which is derived without any non degeneracy condition on the controlled diffusion coefficient. ✷ Proof of Theorem 5.Let us introduce the continuous time version of (5.1

  where E t k ,a[.] denotes the conditional expectation E[.|F t k , I t k = a]. We also denote by E t k [.] the conditional expectation E[.|F t k ]. Since I t k is F t k -measurable, and by the law of iterated conditional expectations, we notice that

	E	sup
		t∈[t k ,t k+1 ]

  Finally, by writing that supt∈[t k ,t k+1 ] |X t -Xπ t k | 2 ≤ 2 sup t∈[t k ,t k+1 ] |X t -X t k | 2 + 2∆X t k, taking successive condition expectations w.r.t to F t ℓ and essential supremum over I t ℓ = a, for ℓ going recursively from k to 0, we get:

	E t k	sup t∈[t k ,t k+1 ]	|X t -Xπ t k | 2 ≤ 2E ess sup a∈A	E t 1 ,a . . . ess sup a∈A	E t k ,a	sup t∈[t k ,t k+1 ]	X t -X t k	2 . . .
			+ 2E ess sup a∈A	E t 1 ,a . . . ess sup a∈A	E t k-1 ,a |∆X t k | 2 . . .
			≤ C 1 |π|,
	by (2.7)-(2.8), which ends the proof.					✷
	3 Discretely jump-constrained BSDE
			a∈A	E t k ,a	sup t∈[t k ,t k+1 ]	X t -X t k	2 . . .	≤ C 1 |π|.	(2.8)

ess sup a∈A E t 1 ,a . . . ess sup

  Proof of Proposition 4.1. The function ϑ is l.s.c. as the supremum of the l.s.c. functions ϑ π . Fix (t, x) ∈ [0, T ) × R d and a, a ′ ∈ A. Let (π p ) p be a sequence of subdivisions of [0, T ] such that |π p | ↓ 0 as p ↑ ∞. We define the sequence (t p ) p of [0, T ] by t p = min s ∈ π p : s > t , p ≥ 0 .

	This proves that
	t k+1
	E
	Combining this last estimate with (4.1) and (4.2), we get the result	✷
			s	x,a	, I t,a s , Ỹs , Zs )ds
	t k+1
	-
		+ E	t	t k+1	f (X t,x,a s	, I t,a s , Ỹs , Zs ) ds .	(4.1)
	From Corollary 3.1, we have
	E|v t k+1					t k+1	1 + |X t,x,a
						t
	By classical estimates, we have
	sup s∈[t,T ]	E |X t,x,a s	| 2 ≤ C(1 + |x| 2 ).
	Moreover, under (H1) and (H2), we know from Proposition 4.2 in [7] that there exists a

s Zs dW s -t k+1 s A Ũs (a ′ )μ(ds, da ′ ) , s ∈ [t, t k+1 ] . From Proposition 3.2, Markov property and uniqueness of a solution to the BSDE (3.5)-(3.6)-(3.7) we have: Ỹs = ϑ π (s, X t,x,a s , I t,a s ), for s ∈ [t, t k+1 ], and so:

|ϑ π (t, x, a) -v π (t k+1 , x)| = Ỹt -v π (t k+1 , x) ≤ E|v π (t k+1 , X t,x,a t k+1 ) -v π (t k+1 , x)| π (t k+1 , X t,x,a t k+1 ) -v π (t k+1 , x)| ≤ C E[|X t,x,a t k+1 -x| 2 ] ≤ C |π| . (4.2)

Moreover, by the growth linear condition on f in (H2), and on ϑ π in Lemma 4.1, we have

E t f (X s , I s , Ỹs , Zs ) ds ≤ CE s | + | Zs | ds . constant C

depending only on the Lipschitz constants of b, σ f and v π (t k+1 , .) such that E sup s∈[t k ,t k+1 ] | Zs | 2 ≤ C(1 + |x| 2 ). t f (X s , I s , Ỹs , Zs ) ds ≤ C(1 + |x|)|π| .

  by the Lipschitz continuity of f . From standard estimate for SDE, we have (recall that the coefficients b and σ are bounded under (H1') and A is compact): for some positive constant C depending only on the Lipschitz constant of f , the upper bound of |f (x, a, 0, 0)| in (H2')(i), and the upper bound of |ϕ ε | in (4.9). Moreover, from the estimate in Proposition 4.2 in[START_REF] Bouchard | Discrete time approximation of decoupled FBSDE with jumps[END_REF] about the coefficients Z ϕε and U ϕε of the BSDE with jumps (4.13), there exists some constant C depending only on the Lipschitz constant of b, σ, f , and of the Lipschitz constant of ϕ ε (t k+1 , .) (which does not depend on ε by (4.11)), such that:

	From standard estimate for the BSDE (4.13), we have:
								E	sup s∈[t,t k+1 ]	|Y ϕε s | 2 ≤ C,
			E	sup
				s∈[t,t k+1 ]
								t,x,a t k+1 ) + E	t	t k+1	f (X t,x,a s	, I t,a s , Y ϕε s )ds
	and so:
		A ε (t, x, a) ≤	1 |π|	E	t	t k+1	f (X t,x,a s	, I t,a s , Y ϕε s ) -f (x, a, ϕ ε (t, x)) ds
			≤ C E	sup s∈[t,t k+1 ]	|X t,x,a s	-x| + |I t,a s -a| + E	sup s∈[t,t k+1 ]	|Y ϕε s -ϕ ε (t, x)| ,
					E	sup s∈[t,t k+1 ]	|X t,x,a s	-x| + |I t,a s -a| ≤ C|π|	1 2 .	(4.14)
	Moreover, by (4.13), the boundedness condition in (H2')(i) together with the Lipschitz
	condition of f , and Burkholder-Davis-Gundy inequality, we have:
	E	sup s∈[t,t k+1 ]	|Y ϕε s -ϕ ε (t, x)| ≤ E |ϕ ε (t k+1 , X t,x,a t k+1 ) -ϕ ε (t, x)|
								+ C|π|E 1 + sup s∈[t,t k+1 ]	|Y ϕε s |
								+ C|π| E	sup s∈[t,t k+1 ]	|Z ϕε s | 2 ] + E	sup s∈[t,t k+1 ] A	|U ϕε s (a)| 2 λ(da)] .

  .1) where ∆t k = t k+1 -t k , E t k [.] stands for E[.|F t k ], and E t k ,a [.] for E[.|F t k , I t k = a]. By induction argument, we easily see that Ȳπ t k is a deterministic function of ( Xπ t k , I t k ), while Ȳ π t k is a deterministic function of Xπ t k , for k = 0, . . . , n, and by the Markov property of the process ( Xπ , I), the conditional expectations in (5.1) can be replaced by the corresponding regressions:

  ). By the martingale representation theorem, there exists Zπ ∈ L 2 (W ) and Ũ π ∈ L 2 (μ) such that and we can then define the continuous-time processes Ȳ π and Ȳπ by: (t k , t k+1 ],for k = 0, . . . , n -1. Denote by δY π t = Y π t -Ȳ π t , δY π t = Y π t -Ȳπ t , δZ π t = Z π t -Zπ t , δU π t = U π t -Ũ π t and δf t = f (X t , I t , Y π t ) -f ( Xπ t k , I t k , Ȳπ t k ) for t ∈ [t k , t k+1). Recalling (3.6) and (5.3), we have by Itô's formula:∆ t := E t k |δY π t | 2 + ∈ [t k , t k+1 ). By the Lipschitz continuity of f in (H2) and Young inequality, we then have:∆ t ≤ E t k |δY π t k+1 | 2 + E t k Xπ t k | 2 + |I s -I t k | 2 + |Y π s -Y π t k | 2 ds .From Gronwall's lemma, and by taking η large enough, this yields for all k ≤ n -1:E t k |δY π t k | 2 ≤ e C|π| E t k |δY π t k+1 | 2 + CB k Xπ t k | 2 + |I s -I t k | 2 + |Y π s -Y π t k | 2 ds ≤ C|π| E t k sup s∈[t k ,t k+1 ] |X s -Xπ t k | 2 + |π|(1 + |X t k |) ,(5.6) by (2.5) and Proposition 3.3. Now, by definition of Y π t k+1 and Ȳ π t k+1 , we have |δY π t k+1 | 2 ≤ ess sup a∈A E t k+1 ,a |δY π t k+1 | 2 . (5.7) By plugging (5.6), (5.7) into (5.5), taking conditional expectation with respect to I t k = a, and taking essential supremum over a, we obtain: ess sup a∈A E t k ,a |δY π t k | 2 ≤ e C|π| ess sup a∈A E t k ,a ess sup a∈A E t k+1 ,a |δY π t k+1 | 2 + C|π| ess sup a∈A E t k ,a sup s∈[t k ,t k+1 ]|X s -Xπ t k | 2 + |π|(1 + |X t k |) .By taking conditional expectation with respect to F t k-1 , and I t k-1 = a, taking essential supremum over a in the above inequality, and iterating this backward procedure until time t 0 = 0, we obtain:E π k (Y) ≤ e C|π| E π k+1 (Y) + C|π| E π k (X) + |π|(1 + E[|X t k |]) ≤ e C|π| E π k+1 (Y) + C|π| 2 , k ≤ n -1, (5.8) where we recall the auxiliary error control E π k (X) on X in (2.2) and its estimate in Lemma 2.1, and set: since g is Lipschitz, and using again the estimate in Lemma 2.1. Observing that E[|δY π t k | 2 ], E[|δY π t k | 2 ] ≤ E π k (Y), we get the estimate: max k≤n E |Y π t k -Ȳ π t k | 2 + E |Y π t k -Ȳπ t k | 2 ≤ C|π|.

	Ȳ π t k+1 = E t k		Ȳ π t k+1 +	t k+1 t k	Zπ t dW t +	t k+1 t k	A	Ũ π t (a)μ(dt, da), k < n,
	Ȳπ t	= Ȳ π t k+1 + (t k+1 -t)f ( Xπ t k , I t k , Ȳπ t k )	(5.3)
		-	t	t k+1	Zπ t dW t -	t	t k+1	A	Ũ π t (a)μ(dt, da),	t ∈ [t k , t k+1 ),
	Ȳ π t	= Ȳ π t k+1 + (t k+1 -t)f ( Xπ t k , I t k , Ȳπ t k )	(5.4)
	-= E t k |δY π t k+1 t t k+1 | 2 + E t k Zπ t dW t -t t ∈ t k+1 t k+1 A Ũ π t (a)μ(dt, da), t |δZ π s | 2 ds + t k+1 t A |δU π s (a)| 2 λ(da)ds t k+1 t 2δY π s δf s ds for all t t k+1 t η|δY π s | 2 ds + C η π|δY π t k | 2 + C η E t k t k+1 where B k = E t k t k+1 t k k (Y) := E ess sup Moreover, by Proposition 3.3, we have sup t∈[t k ,t k+1 ) E |Y π t -Y π t k | 2 + sup This implies finally that: sup s∈(t k ,t k+1 ] E |Y π t -Ȳ π t k+1 | 2 ≤ 2 sup s∈(t k ,t k+1 ] E |Y π t -Y π t k+1 | 2 + 2E |Y π t k+1 -Ȳ π t k+1 | 2 |X s -E π ≤ C|π|,	(5.5)
	as well as
	sup
	s∈[t

t |X s -a∈A E t 1 ,a . . . ess sup a∈A E t k ,a |δY π t k | 2 . . . . By a direct induction on (5.8), and recalling that n|π| is bounded, we get

E π k (Y) ≤ C E π n (Y) + |π| ≤ C(E π n (X) + |π| ≤ C|π|, t∈(t k ,t k+1 ] E |Y π t -Y π t k+1 | 2 ≤ C(1 + E[|X t k |])|π| ≤ C(1 + |X 0 |)|π|. k ,t k+1 ) E |Y π t -Ȳπ t k | 2 ≤ 2 sup s∈[t k ,t k+1 ) E |Y π t -Y π t k | 2 + 2E |Y π t k -Ȳπ t k | 2 ≤ C|π|.

The research of the author benefited from the support of the French ANR research grant LIQUIRISK (ANR-11-JS01-0007).

Approximate optimal control

We now consider the special case where f (x, a) does not depend on y, so that the discrete time scheme (1.4) is an approximation for the value function of the stochastic control problem:

(5.9)

where A is the set of G-adapted control processes α valued in A, and X α is the controlled diffusion in R d :

(Here G = (G t ) 0≤t≤T denotes some filtration under which W is a standard Brownian motion). Let us now define the discrete time version of (5.9). We introduce the set A π of discrete time processes α = (α t k ) k with α t k G t k -measurable, and valued in A. For each α ∈ A π , we consider the controlled discrete time process (X π,α t k ) k of Euler type defined by:

where ∆W t j = W t j+1 -W t j , and the gain functional:

Given any α ∈ A π , we define its continuous time piecewise-constant interpolation α ∈ A by setting: α t = α t k , for t ∈ [t k , t k+1 ) (by misuse of notation, we keep the same notation α for the discrete time and continuous time interpolation). By standard arguments similar to those for Euler scheme of SDE, there exists some positive constant C such that for all α ∈ A π , k ≤ n -1:

from which we easily deduce by Lipschitz property of f and g:

(5.10)

Let us now consider at each time step k ≤ n -1, the function âk (x) which attains the supremum over a ∈ A of θπ k (x, a) in the scheme (5.2), so that:

Let us define the process ( Xπ t k ) k by: Xπ 0 = X 0 , Xπ t k+1 = Xπ