A stability approach for solving multidimensional quadratic BSDEs

We establish an existence and uniqueness result for a class of multidimensional quadratic backward stochastic differential equations (BSDE). This class is characterized by constraints on some uniform a priori estimate on solutions of a sequence of approximated BSDEs. We also present effective examples of applications. Our approach relies on the strategy developed by Briand and Elie in [Stochastic Process. Appl. 123 2921--2939] concerning scalar quadratic BSDEs.


INTRODUCTION
Backward Stochastic Differential Equations Backward stochastic differential equations (BSDEs) have been first introduced in a linear version by Bismut [Bis73], but since the early nineties and the seminal work of Pardoux and Peng [PP90], there has been an increasing interest for these equations due to their wide range of applications in stochastic control, in finance or in the theory of partial differential equations. Let us recall that, solving a BSDE consists in finding an adapted pair of processes (Y, Z), where Y is a R d -valued continuous process and Z is a R d×k -valued progressively measurable process, satisfying the equation (1.1) where W is a k-dimensional Brownian motion with filtration (F t ) t∈R + , ξ is a F T -measurable random variable called the terminal condition, and f is a (possibly random) function called the generator. Since the seminal paper of Pardoux and Peng [PP90] that gives an existence and uniqueness result for BSDEs with a Lipschitz generator, a huge amount of paper deal with extensions and applications. In particular, the class of BSDEs with generators of quadratic growth with respect to the variable z, has received a lot of attention in recent years. Concerning the scalar case,i.e. d = 1, existence and uniqueness of solutions for quadratic BSDEs has been first proved by Kobylanski in [Kob00]. Since then, many authors have worked on this question and the theory is now well understood: we refer to [Kob00,Tev08,BE13] when the terminal condition is bounded and to [BH06,BEK13,DHR11] for the unbounded case. We refer also to [GY14] for a study of BMO properties of Z.
In this paper we will focus on existence and uniqueness results for quadratic BSDEs in the multidimensional setting, i.e. d > 1. Let us remark that, in addition to its intrinsic mathematical interest, this question is important due to many applications of such equations. We can mention for example following applications: nonzero-sum risk-sensitive stochastic differential games in [EKH03,HT16], financial market equilibrium problems for several interacting agents in [ET15,FDR11,Fre14,BLDR15], financial price-impact models in [KP16b,KP16a], principal agent contracting problems with competitive interacting agents in [EP16], stochastic equilibria problems in incomplete financial markets [KXŽ15,XŽ16] or existence of martingales on curved spaces with a prescribed terminal condition [Dar95]. Let us note that moving from the scalar framework to the multidimensional one is quite challenging since tools usually used when d = 1, like monotone convergence or Girsanov transform, can no longer be used when d > 1. Moreover, Frei and dos Reis provide in [FDR11] an example of multidimensional quadratic BSDE with a bounded terminal condition and a very simple generator such that there is no solution to the equation. This informative counterexample show that it is hopeless to try to obtain a direct generalization of the Kobylanski existence and uniqueness theorem in the multidimensional framework or a direct extension of the Pardoux and Peng existence and uniqueness theorem for locally-Lipschitz generators. Nevertheless, we can find in the literature several papers that deal with special cases of multidimensional quadratic BSDEs and we give now a really brief summary of them. First of all, a quite general result was obtain by Tevzadze in [Tev08], when the bounded terminal condition is small enough, by using a fixed-point argument and the theory of BMO martingales. Some generalizations with somewhat more general terminal conditions are considered in [Fre14,KP16a]. In [CN15], Cheridito and Nam treat some quadratic BSDEs with very specific generators. Before these papers, Darling was already able to construct a martingale on a manifold with a prescribed terminal condition by solving a multidimensional quadratic BSDE (see [Dar95]). Its proof relies on a stability result obtained by coupling arguments. Recently, the so-called quadratic diagonal case has been considered by Hu and Tang in [HT16]. To be more precise, they assume that the nth line of the generator has only a quadratic growth with respect to the nth line of Z. This type of assumption allows authors to use Girsanov transforms in their a priori estimates calculations. Some little bit more general assumptions are treated by Jamneshan, Kupper and Luo in [JKL14] (see also [LT15]). Finally, in the very recent paper [XŽ16], Xing and Žitković obtained a general result in a Markovian setting with weak regularity assumptions on the generator and the terminal condition. Instead of assuming some specific hypotheses on the generator, they suppose the existence of a so called Liapounov function which allows to obtain a uniform a priori estimate on some sequence (Y n , Z n ) of approximations of (Y, Z). Their approach relies on analytic methods. We refer to this paper for references on analytic and PDE methods for solving systems of quadratic semilinear parabolic PDEs.
Our approach Our approach for solving multidimensional quadratic BSDEs relies on the theory of BMO martingales and stability results as in [BE13]. To get more into the details about our strategy, let us recall the sketch of the proof used by Briand and Elie in [BE13]. The generator f is assumed to be locally Lipschitz and, to simplify, we assume that it depends only on z. First of all, they consider the following approximated BSDE where ρ M is a projection on the centered Euclidean ball of radius M. Then existence and uniqueness of (Y M , Z M ) is obvious since this new BSDE has a Lipschitz generator. Now, if we assume that ξ is Malliavin differentiable with a bounded Malliavin derivative, they show that Z M is bounded uniformly with respect to M. Thus, (Y M , Z M ) = (Y, Z) for M large enough. Importantly, the uniform bound on Z M is obtained thanks to a uniform (with respect to M) a priori estimate on the BMO norm of the martingale . 0 Z M s dW s . Subsequently, they extend their existence and uniqueness result for a general bounded terminal condition: ξ is approximated by a sequence (ξ n ) n∈N of bounded terminal conditions with bounded Malliavin derivatives and they consider (Y n , Z n ) the solution of the following BSDE Y n t = ξ n + T t f (Z n s )ds − T t Z n s dW s , 0 ≤ t ≤ T, a.s.
By using a stability result for quadratic BSDEs, they show that (Y n , Z n ) is a Cauchy sequence that converges to the solution of the initial BSDE (1.1). Once again, the stability result used by Briand and Elie relies on a uniform (with respect to n) a priori estimate on the BMO norm of the martingale . 0 Z n dW s . The aim of this paper is to adapt this approach in our multidimensional setting. In the first approximation step, we are able to show that Z M is bounded uniformly with respect to M if we have a small enough uniform (with respect to M) a priori estimate on the BMO norm of the martingale  [FDR11]). So, this a priori estimate on the BMO norm of the martingale . 0 Z M s dW s becomes in our paper an a priori assumption and this assumption has to be verified on a case-by-case basis according to the BSDE structure. In the second approximation step, we are facing the same issue: we are able to show the existence and uniqueness of a solution to (1.1) by using a stability result if we have a small enough uniform (with respect to n) a priori estimate on the BMO norm of the martingale . 0 Z n s dW s , and this a priori estimate becomes, once again, an assumption that has to be verified on a case-by-case basis according to the BSDE structure. Let us emphasize that the estimate on the boundedness of Z M and the stability result used in the second step come from an adaptation of results obtained by Delbaen and Tang in [DT08]. The fact that our results are true only when we have a small enough uniform estimate on the BMO norm of the martingale Z n s dW s is the main limitation of our results. Nevertheless, we emphasize that this limitation is related to a crucial open question that could be independently investigated. To be precise, we would like to know if the classical reverse Hölder inequality for exponential of BMO martingales (see Theorem 3.1 in [Kaz94]) stays true in a multidimensional setting, i.e. when we have a matrix valued BMO martingale. For further details we refer the reader to Remark 3.2. To show the interest of these theoretical results, we have to find now some frameworks for which we are able to obtain estimates on the BMO norm of martingales Z n s dW s . This is the purpose of Section 5 where results of [Tev08,Dar95,HT16] are revisited. Let us note that one interest of our strategy comes from the fact that we obtain these estimates by very simple calculations that allow to easily get new results: for example, we are able to extend the result of Tevzadze when the generator satisfies a kind of monotone assumption with respect to y (see subsection 2.2.2). Moreover, we can remark that obtaining such estimates is strongly related to finding a so-called Liapounov function in [XŽ16]. Result on the boundedness of Z is also interesting in itself since it allows to consider the initial quadratic BSDE (1.1) as a simple Lipschitz one which gives access to numerous results on Lipschitz BSDEs: numerical approximation schemes, differentiability, stability, and so on.
Structure of the paper In the remaining of the introduction, we introduce notations, the framework and general assumptions. We have collected in Section 2 all our main results in order to improve the readability of the paper. Section 3 contains some general results about SDEs and linear BSDEs adapted from [DT08]. Section 4 is devoted to the proof of stability properties, existence and uniqueness theorems for multidimensional quadratic BSDEs. Finally, proofs of the applications of previous theoretical results are given in Section 5. § 1.1. Notations ⋄ Let T > 0. We consider Ω, F , (F t ) t∈[0,T ] , P a complete probability space where (F t ) t∈[0,T ] is a Brownian filtration satisfying the usual conditions. In particular every càdlàg process has a continuous version. Every Brownian motion will be considered relatively to this filtered probability space. A k-dimensional Brownian motion W = W i 1 i k is a process with values in R k and with independent Brownian components. Almost every process will be defined on a finite horizon [0, T ], either we will precise it explicitly. The stochastic integral of an adapted process H will be denoted by H ⋆ W , and the Euclidean quadratic variation by ., . . The Dolean-Dade exponential of a continuous real local martingale M is denoted by ⋄ Linear notions -On each R p , the scalar product will be simply denoted by a dot, including the canonical scalar product on M dk (R): for all x ∈ R d×k . If A and B are two processes with values in M dk (R) and R k , the quadratic variation A, B is the R d vector process and we have the integration by part formula d (AB) = dA.B + A. dB + d A, B . We can also define the covariation of (A, .
⋄ Functional spaces -In a general way, Euclidean norms will be denoted by |.| while norms relatively to ω and t will be denoted by . . For a F -adapted continuous process Y with values in R d and 1 p ∞ , let us define If Z is a random variable with values in R d , we define In the sequel, to simplify notations we will skip the superscript . 1 on the Brownian motion after a star. For more details about BMO martingales, we can refer to [Kaz94].
there exists a sequence (c n ) of positive constants, such that, for all n ∈ N, where B n (b 0 ) states for the Euclidean ball on R d of center b 0 and radius n. If the last term does not depend on b 0 , we shall say that g is in C (α n ),loc . Finally, for a given α ∈ (0, 1], a function g : Remark -1.1. We can plainly show that if there exists (α n ) ∈ (0, 1] N such that a bounded solution v is in C (α n ),loc , then v ∈ C α 1 .
⋄ Inequalities -BDG inequalities claim that . S p and . H p are equivalent on martingale spaces with two universal constants denoted C ′ p ,C p . It means that for all continuous local martingales M vanishing at 0, Between τ and σ , the started and stopped process is simply a translation of the stopped process: for all u such that τ u σ a.s, τ X ⌋σ u = X u − X τ . This process is constant after σ and vanishes before τ. Let us suppose that X is a BMO martingale. We say that X is ε-sliceable if there exists a subsequence of stopping times 0 = T 0 T 1 ...
The set of all ε-sliceable processes will be denoted by BMO ε . Schachermayer proved in [Sch96] that Moreover the BMO norm of a started and stopped stochastic integral process τ Z ⋆W ⌋σ has a simple expression: where T τ,σ = τ ′ stopping time : τ τ ′ σ a.s .
A proof of this proposition is given in the appendix part. ⋄ Malliavin calculus -We denote by the set of all Wiener functions. For F ∈ P, the Malliavin derivative of F is a progressively measurable In particular D ((h ⋆ W ) T ) = h for all adapted process h. We define a kind of Sobolev norm on P with the following definition We can show that D is closable, consequently it is possible to extend the definition of D to D 1,2 = P 1,2 .
Besides, D 1,2 is dense in L 2 (Ω). For further considerations on Malliavin calculus we can refer to [Nua06]. We finish this paragraph by the following useful result proved in [Nua06] (Proposition 1.2.4).
Proposition -1.4. Let ϕ : R d → R. We assume that there exists a constant K such that for all x, y ∈ R d , . Then ϕ(F) ∈ D 1,2 (R d ) and there exists a random vector (G 1 , ..., G d ) such that G i DF i , and |G| K. § 1.2. Framework and first assumptions In this paper we consider the following quadratic BSDE on R d : (1.5) where f is a random function Ω × [0, T ] × R d × R d×k → R d called the generator of the BSDE such that for all (y, z) ∈ R d × R d×k and t ∈ [0, T ], ( f (t, y, z)) 0 t T is progressively measurable, (Y, Z) is a process with values in R d × R d×k and ξ ∈ L 2 F T , R d .
Definition -1.1. A solution of BSDE (1.5) is a process (Y, Z) ∈ S 2 (R d ) × H 2 (R d×k ) satisfying usual integrability conditions and solving initial BSDE: Some locally Lipschitz assumptions on f and integrability assumptions on ξ and f will be assumed all along this paper.
(H) (i) For all (y, y ′ , z, z ′ ) ∈ R d 2 × R d×k 2 , we assume that there exists (K y , L y , K z , L z ) ∈ (R + ) 4 such that P − a.s for all t ∈ [0, T ]: We denote by B m (L y , L z ) the following quantity depending on L y and L z : (1.6) For all m > 1, let us denote by Z m BMO the set which can be rewritten as where B m (L y , L z ) is defined in (1.6). We also denote by Z slic,m BMO the set of all R d×k -valued processes Z for which there exists a sequence of stopping times 0 = T 0 T 1 ... T N = T such that T i Z ⌋T i+1 ∈ Z m BMO for all i ∈ {0, ..., N}.
To conclude this introduction, we finally consider an approximation of the BSDE (1.5). To this purpose let us introduce a localisation of f defined by f M (t, y, z) = f (t, y, ρ M (z)) where ρ M : R d×k → R d×k satisfies the following properties : Thus f M is a globally Lipschitz function with constants depending on M. Indeed we have for all (t, y, y ′ , z, z ′ ) ∈ Then, according to the classical result of Pardoux and Peng in [PP90], there exists a unique solution (1.7)

MAIN RESULTS
We have collect in this section principal results proved in our article. All proofs are postponed to sections 4 and 5. The following subsection gives some existence and uniqueness results while subsection 2.2 is dedicated to particular frameworks where these existence and uniqueness results apply. § 2.1. Some general existence and uniqueness results 2.1.1 Existence and uniqueness results when the terminal condition and the generator have bounded Malliavin derivatives We consider here a particular framework where the terminal condition and the random part of the generator have bounded Malliavin derivatives. More precisely, let us consider the following assumptions.
By recalling that (Y M , Z M ) is the unique solution of (1.7), we will also assume that we have an a priori estimate on |Z M | ⋆ W uniform in M and small enough. For a given m > 1 we consider the following assumption: Theorem -2.1 (Existence and uniqueness (1)). A result similar to Theorem -2.1 can be obtained when the quadratic growth of z has essentially a diagonal structure. Thus, we replace assumption (H) by the following one: (Hdiag) • There exist f diag : • There exist five nonnegative constants L d , K d,y , L d,y , K d,z , L d,z such that for all (t, y, y ′ , z, This kind of framework has been introduced by Hu and Tang in [HT16] (see also [JKL14]). The following result of existence and uniqueness is specific, and do not follows directly from Theorem -2.1. Indeed, if an uniform upper bound is assumed (assumption (i) below), we can use specific tools in the diagonal case to obtain an upper bound small enough.
Theorem -2.2 (Existence and uniqueness (1) -Diagonal Case). We assume that (Hdiag), (Dxi,b), (Df,b) hold true and that there exists a constant B such that where c 1 and c 2 are given by Proposition -1.1 with B = 2L d B.
We also assume that ξ ∈ L ∞ (Ω, F T ) and f (., 0, 0) ∈ S ∞ (R d ). Then, the quadratic BSDE (1.5) has a unique solution The main difference between assumptions in Theorem 2.1 and Theorem 2.2 comes from the form of constants used in the bound of the BMO norm. In particular, for any L d > 0, there exists ε > 0 such that (ii) in Theorem 2.2 is fulfilled as soon as L d,y < ε and L d,z < ε while we cannot take L z as large as we want in Theorem 2.1.

Extension to general terminal values and generators
Now we are able to relax assumptions (Dxi,b) and (Df,b) with some density arguments. To do so, we assume that we can write f as a deterministic function f of a progressively measurable continuous process: the randomness of the generator will be contained into this process.
Besides, we assume that (H) holds true for f.
. Finally, assumption (BMO,m) will be replaced by the following one.
• Let us emphasize that the uniqueness result in Theorem 2.3 lies in a different space than the space used in Theorem 2.1.
• It is also possible to extend the result of Theorem -2.2 (diagonal case) to more general terminal conditions and generators. Nevertheless, the result obtained would be less general than Theorem -2.3. See Remark -4.8 for more details. § 2.2. Applications to multidimensional quadratic BSDEs with special structures In this subsection we give some explicit frameworks where assumptions (BMO,m) and (BMO2,m) or assumptions (i) and (ii) of Theorem -2.2 are fulfilled. The aim is to show that numerous results on multidimensional quadratic BSDEs already proved in the literature can be obtained with similar assumptions by our approach. We want to underline the simplicity of this approach since we just have to obtain some a priori estimates on the BMO norm of |Z| ⋆W by using classical tools as explained in section 5. Moreover, it is quite easy to construct some « new » frameworks where (BMO,m) and (BMO2,m) or assumptions (i) and (ii) of Theorem -2.2 are also fulfilled.

An existence and uniqueness result for BSDEs with a small terminal condition
In [Tev08], Tevzadze obtains a result of existence and uniqueness for multidimensional quadratic BSDEs when the terminal condition is small enough by using a contraction argument in S ∞ × BMO. We are able to deal with this kind of assumption with our approach. We consider the following hypothesis.

An existence and uniqueness result for BSDEs with a monotone generator
In this part we investigate the case where we have for f a kind of monotonicity assumption with respect to y.

An existence and uniqueness result for diagonal quadratic BSDEs
Now we consider the diagonal framework introduced in section 2.1.1. We assume that the generator satisfies (Hdiag), i.e. the generator f can be written as f (t, y, z) = f diag (t, z) + g(t, y, z) where f diag has a diagonal structure with respect to z.
Proposition -2.3. We assume that where c 1 and c 2 are given by Proposition -1.
Remark -2.2. The growing assumption (2.3) is only one example of hypothesis that can be tackled by our approach. It is also possible to obtain the same kind of result by replacing (2.3) by one of the following assumption: • We assume that for all (t, y, z) |g(t, y, z)| C(1 + |y|) + ε |z| 2 and T, ε are supposed to be small enough. This framework is studied in [HT16,JKL14].
This situation is already studied in [HT16].

Existence and uniqueness of martingales in manifolds with prescribed terminal condition
The problem of finding martingales on a manifold with prescribed terminal value has generated a huge amount of literature. On the one hand with geometrical methods, Kendall in [Ken90] treats the case where the terminal value lies in a geodesic ball and is expressed as a functional of the Brownian motion. Kendall gives also a characterisation of the uniqueness in terms of existence of a convex separative function, i.e. a convex function on the product space which vanishes exactly on the diagonal. Besides, in [Ken92], Kendall proved that the property every couple of points are connected by a unique geodesic is not sufficient to ensure existence of a separative convex function, which was conjectured by Émery. An approach by barycenters, of the martingale notion on a manifold, is used by Picard in [Pic94] for Brownian filtrations. Arnaudon in [Arn97] solved the problem in a complex analytic manifold having a convex geometry property for continuous filtrations: the main idea is to consider a differentiable family of martingales. For all these results, a convex geometry property is assumed. The first approach using the tool of BSDEs is proposed by Darling in [Dar95]. Let us now define more precisely the problem. A so-called linear connection structure is required to define martingales on a manifold M in a intrinsic way. A contrario, for semimartingales, a differential structure is enough. The definition of a martingale can be rewritten with a system of coupled BSDEs having a quadratic growth, so we begin to recall it. We can refer to [Eme89] for more details about stochastic calculus on manifolds. Let us consider (M , ∇) a differential manifold equipped with a linear connection ∇. This is equivalent to give ourselves a Hessian notion or a covariant derivative. We say that a continuous process X is a semimartingale on M if for all F ∈ C 2 (M ), F • X is a real semimartingale. Consistence of the definition is simply due to the Itô formula. We say that a continuous process Y is a (local) ∇-martingale if for all F ∈ C 2 (M ), Again it is not very hard to see with the Itô formula that this definition is equivalent to the Euclidean one in the flat case. Let us remember that . 0 ∇ dF(dY, dY ) s is a notation for the quadratic variation of Y with respect to the (0, 2)-tensor field ∇ dF. This notion is defined by considering a proper embedding ( On the other hand it can be proved that the quantity The coefficients are symmetric with respect to i, j. Hence martingale property in the domain of a local chart is equivalent to the existence of a process Z such that (Y, Z) solves the following BSDE It is an easy consequence of the representation theorem for Brownian martingales and the definition applied to F = x i . We consider in addition the following assumption (HGam) there exists two constants L y and L z such that for all i, j, k ∈ {1, ..., d} For example (HGam) is in force if the domain of the chart is a compact set. It is also true if we choose an exponential chart. Without loss of generality we can suppose that M has a global system of coordinates: all the Christoffel symbols will be computed in this system.
and with the symmetric property of the Christoffel symbols, we have which implies that To obtain some important a priori estimate for the BMO norm of Z ⋆W , Darling introduce in [Dar95] a convex geometry assumption.
Definition -2.1. We say that a function F ∈ C 2 (M , R) (seen as a function on This property means that F is convex with respect to the flat connection, and, with respect to the connection ∇.
Theorem -2.4. Let m > 1 and assume that: (ii) F dc is doubly convex on M , and there exists α > 0 and m 1 such that F dc is α-strictly doubly convex on G and satisfies Remark -2.3. By using the same approach, it should be possible to extend the previous result to ∇-Christoffel symbols that depend on time or even that are progressively measurable random processes.

The Markovian setting
The aim of this subsection is to refine some results of Xing and Žitković obtained in [XŽ16]: in this paper, authors establish existence and uniqueness results for a general class of Markovian multidimensional quadratic BSDEs. Let us start by introducing the Markovian framework. For all t ∈ [0, T ] and x ∈ R k we denote X t,x a diffusion process satisfying the following SDE (2.4) In all this part, we assume following assumptions that ensure, in particular, that for all (t, x) ∈ [0, T ] × R k , there exists a unique strong solution of (2.4). (HX) The aim of this subsection is to study the following Markovian BSDE for which we assume following assumptions: (HMark) As in [XŽ16] we say that a pair (v, w) of functions is a continuous Markovian solution of (2.5) if ) is a solution of (2.5).
Some existence and uniqueness results about continuous Markovian solutions of (2.5) are obtained in [XŽ16] by assuming the existence of a so-called Lyapunov function. We recall here the definition of these functions given in [XŽ16].
We are now able to give a uniqueness result that partially refine the result given by [XŽ16].
Theorem -2.5 (Uniqueness for the Markovian case). We assume that (i) (HX) and (HMark) are in force.
(ii) there exists a Lyapunov function F associated to f.
Then (2.5) admits at most one continuous Markovian solution (v, w) such that v is bounded.
Moreover, we are also able to precise the regularity of the solution when it exists.
Theorem -2.6 (Regularity of the Markovian solution). We assume that: (i) (HX) and (HMark) are in force, (ii) there exists D ∈ R + and κ ∈ (0, 1] (same constant κ as in (HMark)) such that for all (s, (iii) there exists a Lyapunov function F associated to f.
If (v, w) is a continuous Markovian solution of (2.5) such that v is bounded, then v ∈ C κ . Particularly, if κ = 1 then w is essentially bounded: the multidimensional quadratic BSDE (2.5) becomes a standard multidimensional Lipschitz BSDE by a localisation argument.
Remark -2.4. An existence result is given by Theorem 2.7 in [XŽ16]. A less general existence result can be obtained thanks to our approach by combining estimates obtained by Xing and Žitković in Theorem 2.5 of [XŽ16], small BMO estimates obtained in the proof of Theorem -2.5 and Remark -4.7 but the approach is less direct than in [XŽ16]. Concerning the uniqueness, Xing and Žitković have proved a uniqueness result for generators that do not depend on y: our result allows to fill this small gap. Finally, Xing and Žitković prove that there exists a Markovian solution that satisfies v ∈ C κ ′ ,loc with κ ′ ∈ (0, κ]. Thus, our regularity result gives a better estimation of the solution regularity since the regularity of the terminal condition and the generator is retained. In particular, we obtain that Z is bounded when κ = 1 which can have important applications, as pointed out in the introduction. Remark -2.5. The existence of a Lyapunov function seems to be an ad hoc theoretical assumption at first sight but Xing and Žitković provide in [XŽ16] a lot of examples and concrete criteria to obtain such kind of functions. Moreover we can note that the Lyapunov function can be used to obtain a priori estimates on |Z| ⋆ W BMO (see the proof of Theorem -2.5 and Theorem -2.6).

GENERALITIES ABOUT SDES AND LINEAR BSDES
We collect in this section some technical results that will be useful for section 4 and section 5. § 3.1. The linear case: representation of the solutions We investigate here the following linear BSDE For the linear case we have an explicit formulation of the solution. Let us begin to recall the classical scalar formula which can be obtained using the Girsanov transform.
Remark -3.1 (One-dimensional case (d = 1)). It is well-known that the solution of (3.1) is given by the formula To extend this last formula in the general case we define, as in [DT08], a process S as the unique strong solution of Proposition -3.1 (Formula for U). (ii) The BSDE (3.1) has a unique solution (U,V ) in S 2 R d × H 2 R d×k , and U is given by: Proof. Existence and uniqueness of a solution (U,V ) in S 2 (R d ) × H 2 (R d×k ) is guaranteed by the Pardoux and Peng result in [PP90]. The solution (U,V ) satisfies The Itô formula gives the invertibility of S and the formula for S −1 on the one hand. On the other hand: and thus we get, for all t ∈ [0, T ],

A result about SDEs
We consider a SDE on R d×d of the form Proposition -3.2. Let m 1. We suppose that there are two non-negative adapted processes α and β such that Then there exists a solution X ∈ S m (R d ) to the equation (3.3) and a constant K m,ε 1 ,ε 2 such that For the reader convenience a proof of this result can be found in the appendix. From this last proposition we can deduce the following corollary (see [DT08], Corollary 2.1) Corollary -3.1. Let m 1. We suppose that there are two non-negative adapted processes α and β such that Then we get, for all t ∈ [0, T ], Finally, the definition of conditional expectation gives us the result. If X is invertible and if F and G are linear with respect to x, the process X −1 t X is for all t a solution taking the value I d at s = t. The particular case is shown using (3.4).
Remark -3.2. The main limitation of Corollary -3.1 comes from assumption (ii): we need to have a small BMO norm estimate on processes ( √ α ⋆ W , β ⋆ W ) to get a reverse Hölder inequality. It is well known that we have a more general result when d = 1: if α = 0 and β ⋆ W ∈ BMO then there exists m > 1 (that depends on the BMO norm of β ⋆ W ∈ BMO) such that X satisfies a reverse Hölder inequality with the exponent m (see Theorem 3.1 in [Kaz94] and references inside, or [CM13] for a new recent proof). We do not know if this result stays true in the multidimensional framework but we emphasize that this is a crucial open question. Indeed, if such a result is true, then we whould be able to prove that Theorem -2.1 and Theorem -2.3 stay true without assuming K < B m (L y , L z ) in hypothesis (BMO,m) (at least when the generator is Lipschitz with respect to y, i.e. L y = 0). § 3.3. Estimates for the solution to BSDE (3.1) We come back to the linear BSDE (3.1), and we want to obtain some S q -estimations for U with q large enough, including q = ∞, under BMO assumptions.
Proposition -3.3. Let m 1. We assume that B and A are adapted, bounded respectively by two non negative processes β and α such that: Then (i) If ζ ∈ L ∞ (Ω, F T ) and f ∈ S ∞ , then U ∈ S ∞ (R d ) and In the following we will denote simply K q,m,ε 1 ,ε 2 = 2 q−1 K q m,ε 1 ,ε 2 q q − m * q/m * . Proof. The formula (3.2) gives us, for all t ∈ [0, T ]: S is the solution of an SDE on R d×d for which we can use Corollary -3.1 by taking, for all 1 p k and (x, y) ∈ (R d×d ) 2 , G p (s, x) = xB (:,p,:) s and F(s, y) = yA s . Let us note that B (:,p,:) |B| for all p ∈ {1, ..., k}. Thus there exists a constant K m,ε 1 ,ε 2 such that: ⋄ If ζ ∈ L ∞ and f ∈ S ∞ , by using the Hölder inequality we have ⋄ Let us consider m > 1 and assume that ζ ∈ L ∞ , | f | ⋆ W is BMO. Then, by using Hölder and energy inequalities ⋄ Let us consider m > 1 and q > m * . We get, for all t ∈ [0, T ], So we obtain the announced result: Corollary -3.2 (Affine upper bound). Let m 1. Let us consider A and B adapted, bounded respectively by two real processes α and β of the form We have the following estimates, with constants K m , K q,m depending only on m, q, K y , K z , L y , L z and the BMO In the following we will denote simply K q,m = 2 q−1 K q m q q − m * q/m * . Proof. We obtain easily estimates about BMO-norms of √ α ⋆ W and β ⋆ W by using the triangle inequality, and it follows that √ α ⋆ W, β ⋆ W are BMO. To use Proposition -3.3 we just have to show that √ α ⋆ W and β ⋆ W are respectively ε 1 and ε 2 sliceable with 2mε 2 1 + √ 2ε 2 C ′ m < 1. To this end, we consider the following uniform sequence of deterministic stopping times and a parameter η > 0. With Proposition -1.3 and defining η = T N , previous inequalities become on (3.6) By taking η small enough, we get 2mε 2 1 + √ 2ε 2 C ′ m < 1 since the following upper bound holds true Remark -3.3. In inequalities (3.5) and (3.6), we have used that We can easily obtain a more general result by replacing the following assumption: A , B are two positive real processes such that by the new one: A , B are two positive real processes such that √ A ⋆ W, B ⋆ W are in BMO ε 1 and BMO ε 2 with the condition 2mLε 2 1 + √ 2L ′ ε 2 C ′ m < 1.

Proof.
Step 1 -Malliavin differentiation. We assume that f is continuously differentiable with respect to (y, z). This assumption is not restrictive by considering a smooth regularization of f . Recalling assumptions (Dxi,b) and (Df,b), Proposition 5.3 in [EKPQ97] gives us that for all 0 u t T , Y M t and Z M t are respectively in D 1,2 (R d ) and D 1,2 (R d×k ). Moreover the process and (D t Y t ) 0 t T is a version of (Z t ) 0 t T . In particular, there exists a continuous version of Z. Let us empha-size that BSDE (4.1) means that for each p ∈ {1, ..., k}, besides D p Y M is a process with values in R d for each p ∈ {1, ..., k}.
Step 2 -S ∞ -Estimation. We are looking for an S ∞ -estimate of D u Y M for all u ∈ [0, T ] applying results of section 3. Since |∇ z ρ M (z)| 1, we obtain the following inequalities by recalling the main assumption (H), Let us consider the two positive processes α M and β M defined below, For all p ∈ {1, ..., k}, by recalling (BMO,m), we can apply Corollary -3.2 (iii), to the BSDE (4.2) with the following constants and processes: Thus, we obtain, for all u ∈ [0, T ], where C m does not depend on M. Indeed, it is important to remark that the constant K m given by Corollary -3.2 depends on Z M ⋆ W BMO and so, could depend on M. But, by checking the proof of Proposition -3.2 in the Appendix it is easy to see that the constant K m given by Corollary -3.2 is equal to where N is an integer large enough and the uniform bound with respect to M follows. Under the assumption (Df,b) together with (BMO,m), the last term has a S ∞ -upper bound uniform with respect to M. Indeed we have, for all (u,t) ∈ [0, T ] 2 , The last supremum is finite under assumption (BMO,m) and we obtain the announced result since When f is not continuously differentiable with respect to (y, z) we consider a smooth regularization of f and we obtain by this classical approximation that We are now able to prove Theorem -2. Thanks to assumptions on f and f M , we get Then, Y M ⋆ , Z M ⋆ becomes a solution of the quadratic BSDE (1. In this case, if all the other assumptions of Theorem -2.1 are fulfilled, then the quadratic BSDE (1.5) has a unique solution (Y, Z) ∈ S ∞ (R d ) × Z slic,m BMO such that esssup Ω×[0,T ] |Z| < +∞.
We do not give the proof of Theorem -2.2 since it is quite similar to the proof of Theorem -2.1. Indeed, the main point is to show that Proposition -4.1 stays true. To do that we just have to mimic its proof and replace the application of Corollary -3.2 (iii) by a new tailored one adapted to the diagonal framework and proved by using the same strategy as in the proof of Theorem -4.2. § 4.2. Stability result With the classical linearisation tool we can prove a stability theorem for the BSDE (1.5) by using results of section 3. Let us consider two solutions of (1.5) in R d × R d×k , denoted (Y 1 , Z 1 ) and (Y 2 , Z 2 ), with terminal conditions ξ 1 and ξ 2 and generators respectively f 1 and f 2 : We assume that f 1 , f 2 satisfies the usual conditions (H). Let us denote s , Z 2 s ) and δ ξ = ξ 1 − ξ 2 . The process (δY, δ Z) solves the BSDE Theorem -4.1 (Stability result). Let m > 1, p > m * 2 and let us suppose that Then, there exists a constant K p Z 1 ⋆ W BMO , Z 2 ⋆ W BMO (depending only on p, K y , L y , K z , L z , T and the BMO norms of Z 1 ⋆ W and Z 2 ⋆ W ) such that Proof. We firstly assume that By using the classical linearisation tool, we can rewrite (4.3) as where ⋄ B is a L (R d×k , R d ) process defined by blocks by, for all i ∈ {1, ..., k}, Assumption (H) on f 1 and f 2 gives the following inequalities: Step 1 -Control of δY . A and B are bounded respectively by two real processes α and β defined by and (δY, δ Z) solves a linear BSDE of the form (3.1) with δ f instead of f . We can apply Corollary -3.2, (iii) with B = Z 1 + Z 2 , A = Z 1 2 , L ′ = L z , K = K y , K ′ = K z , and L = L y , which gives, for all q > 1 such that q > m * , (4.4) Step 2 -Control of δ Z. The Itô formula applied to |δY | 2 gives us Recalling assumption (H) we have With the Cauchy-Schwarz and Young inequalities, we get By using this last inequality in (4.5) we obtain Thus, for all p 1, there exists a constant K depending only on p such that In the following we keep the notation K for all constants appearing in the upper bounds. Then, according to the BDG inequalities, we get for all p 1: , then the Cauchy-Schwartz inequality gives us Moreover we obtain with Cauchy-Schwarz and Young inequalities: The energy inequality allows us to bound which is finite recalling assumption (i). Finally, for all p 1, there exists a constant K (which depends only on p, K y , L y , K z , L z , T and the BMO norms of (4.6) Step 3 -Stability. Considering p > m * 2 and combining (4.4) where q = 2p with (4.6), we obtain existence of a constant K p Z 1 ⋆ W BMO , Z 2 ⋆ W BMO which depends only on p, K y , L y , K z , L z , T, K and the BMO norms of Z 1 ⋆ W , Z 2 ⋆ W such that .
Step 4 -The alternative assumption (i) We can deal with assumption (Z 1 , Z 2 ) ∈ Z m BMO × Z m BMO by changing the linearization step in the proof. We can remark that δ F s = A s δY s + B s δ Z s + δ f s , where and we get symmetric bounds for A and B: Then (i) becomes wich is fulfilled as soon as we have Remark -4.2. A more restrictive stability result is already obtained in [KP16b] (see Theorem 2.1).
Remark -4.3. By using Remark -3.3, it is clear that Theorem -4.1 stays true when Z 1 and Z 2 are only in Z slic,m BMO . Indeed, if we denote 0 = T j 0 T j 1 ... T j N j the sequence of stopping times associated to Z j ⋆W for j ∈ {1, 2}, we can define a new common sequence of stopping times: Then, by applying the stability result on each interval Obviously, when sequences of stopping times are the same for Z 1 ⋆W and Z 2 ⋆W , we can use it directly as the common sequence of stopping time. § 4.3. Stability result for the diagonal quadratic case We give here a specific result when the quadratic growth of z has essentially a diagonal structure: we assume that assumption (Hdiag) is in force. As explained in section 2.1.1, this kind of framework has been introduced by Hu and Tang in [HT16] (see also [JKL14]).
To simplify notations in this paragraph, the line i of z will be denoted in a simple way by (z) i , or z i if there is no ambiguity, instead of z (i,:) . Let us consider two solutions (Y 1 , Z 1 ) and (Y 2 , Z 2 ) which correspond to terminal conditions ξ 1 , ξ 2 and generators f 1 = f diag,1 + g 1 , f 2 = f diag,2 + g 2 . We have for all i ∈ {1, ..., d}, We also define

Theorem -4.2 (Stability result for the diagonal quadratic case). Let us assume that
(i) f 1 and f 2 satisfy (Hdiag), where c 1 and c 2 are given by Proposition -1.1 with B = 2L d B.
Then there exists a constant K diag Z 1 ⋆ W BMO , Z 2 ⋆ W BMO depending only on B and constants in (Hdiag) such that Remark -4.4. For a given L d and a given B, condition (4.7) is fulfilled as soon as L d,y and L d,z are small enough. Proof.
Step 1 -Control of δY . We write δ F i as where β i , α i and γ i are defined by: Since we have the following estimate on β i , for all i ∈ {1, ..., d}, Consequently we can apply the Girsanov theorem: where W i is a Brownian motion with respect to the probability Q i defined by dQ i = E β i ⋆ W T dP. Taking the Q i -conditional expectation we get Following estimates hold true: and consequently, we obtain Since (|α i | |δY | + |γ i | |δ Z| + |δ f i |) ⋆W is a BMO martingale, we can apply Proposition -1.1: there exists a constant c 2 that depend only on L d and B such that and consequently we get As in the proof of Proposition -3.2, now we slice [0, T ] in small pieces. We consider η = T N with N ∈ N * and On the interval [T k , T k+1 ] the inequality (4.9) becomes: Then, we can choose N large enough to get 1 − c 2 2 d(K d,y η + L d,y B 2 ) > 0. Finally we obtain . (4.11) Step 2 -Control of δ Z. Applying the Itô formula for the process δY i 2 and taking the Q i -conditional expectation, we get for all t ∈ [0, T ], Martingales By using Proposition -1.1, there exist two constants c 1 > 0 and c 2 > 0 that depend only on L d and B such that By summing with respect to i and by using assumption (Hdiag) we obtain Once again, for each k ∈ {0, ..., N − 1} we can write this inequality on [T k , T k+1 ], and with the same notations as in (4.10) we obtain We apply Young inequality to the terms T k |δ Z| ⋆ W ⌋T k+1 BMO and |δ f | ⋆ W 2 BMO and we obtain and for all ε > 0 Consequently we get (4.12) Step 3 -Stability. Combining (4.11) and (4.12), we can obtain a stability result on [T k , T k+1 ] as soon as η and ε are sufficiently small to get c 2 2 d(K d,y η + L d,y B 2 ) < 1 and √ 2 We obtain the existence of a constant K which does not depend on k such that , by a direct iteration we finally obtain a constant K such that and K depends only on B and constants in (Hdiag). § 4.4. Proof of Theorem -2.3 Theorem -2.3 is proved by relaxing assumptions (Dxi,b) and (Df,b) of Theorem -2.1 thanks to some density arguments. To ensure the convergence, the keystone result will be the stability Theorem -4.1.

Proof. [of Theorem -2.3]
Step 1-Approximations. We can approach ξ with a sequence of random variables (ξ n ) n∈N such that for every n, ξ n has a bounded Malliavin derivative: More precisely ξ n can be chosen of the form Φ n (W t 1 , ...,W t n ) where Φ n ∈ C ∞ b (R n ), (t 1 , ...,t n ) ∈ [0, T ] n and ξ n tends to ξ in every L p for p 1 (see [Nua06], Exercise 1.1.7). Since α is adapted, we can approach this process with a sequence of sample processes α n of the form where (t n i ) p n i=0 is a sequence of subdivisions of [0, T ], with sup 0 i p n −1 t n i+1 − t n i −→ n→∞ 0, and, for all 0 i p n − 1, n ∈ N, α i,n is a F t n i -measurable random variable. We have a convergence of this sequence to α in L 2 (Ω × [0, T ]): We can assume in addition that for all n and for all 0 i p n , α i,n has a bounded Malliavin derivative since this set is dense in L 2 (Ω). It is obvious that for all 0 u T and 0 t T , According to Proposition -1.4 applied to ϕ = f(., y, z), there exists for all n ∈ N and t ∈ [0, T ] a bounded random variable G such that D t f(α n t , y, z) = G.D t α n t , and |G| D(1 + |z| 2 ).
For each n ∈ N: ξ n satisfies (Dxi,b), f(α n . , ., .) satisfies (Df,b) and (BMO,m) is fulfilled. So, we can apply Theorem -2.1: there exists an unique solution (Y n , Z n ) ∈ S 2 (R d ) × Z m BMO of the equation Step 2-Application of the stability result. We can assume that for all n, ξ n L 2m * ξ L 2m * . If it is not true, we consider the sequence ξ n = ξ L 2m * ξ L 2m * + ξ n − ξ L 2m * ξ n instead of ξ n . The same argument allows us to assume that α n Under (BMO2,m), we have the estimate Hence, for all n 1 , n 2 ∈ N, we can use Theorem -4.1 for p = m * which gives us: where the constant K m * appearing does not depend on n under (BMO2,m). This fact was already highlighted in the proof of Proposition -4.1 where an explicit formula for K m * was given. We recall that (ξ n ) n∈N is a Cauchy sequence in L 2m * , so ξ n 1 − ξ n 2 L 2m * −→ n 1 ,n 2 →∞ 0.
For the second term, we use the Hölder inequality: then, a uniform integrability argument gives us Finally we get With the energy inequality, the first term is uniformly bounded with respect to n 2 under the assumption (BMO2,m) by choosing 1 < p 2 2 − η . The second one tends to zero when n 1 , n 2 go to infinity since the convergence in every H r for r > 1 holds true.
Remark -4.7. By using Remark -3.3 once again, Theorem -2.3 can be adapted if we replace the assumption (BMO2,m) by the following one: ξ ∈ L 2m * and there exist a constant K and a sequence 0 = T 0 T 1 ...
T N = T of stopping times (that does not depend on M) such that In this case, if all the other assumptions of Theorem -2.3 are fulfilled, then the quadratic BSDE (1.5) has a unique solution (Y, Z) ∈ S ∞ (R d ) × Z slic,m BMO such that esssup Ω×[0,T ] |Z| < +∞.
Remark -4.8. It is possible to extend the existence and uniqueness result for the diagonal case given by Theorem -2.2 to more general terminal conditions and generators. More precisely, it is possible to apply the same strategy as for the proof of Theorem -2.3 by applying the stability result given by Theorem -4.2 instead of Theorem -4.1. Nevertheless we can only obtain an existence and uniqueness result for terminal conditions (resp. generators) that can be approximated in L ∞ (resp. BMO) by terminal conditions satisfying assumption (Dxi,b) (resp. generators satisfying assumption (Df,b)).

BMO .
We can note that upper bound do not depend on M.
Proof. To simplify notations in the proof, we skip the superscript M on (Y M , Z M ) and f M . The unique solution (Y, Z) ∈ S 2 × H 2 of (1.7) can be constructed with a Picard principle as in the seminal paper of Pardoux and Peng (see [PP90]). We consider a sequence (Y n , Z n ) n∈N such that (Y n , Z n ) n∈N tends to (Y, Z) in S 2 R d × H 2 R d×k . This sequence is given by We will prove with an induction that: for all n ∈ N, Y n ∈ S ∞ , |Z n | ⋆ W ∈ BMO and The case n = 0 is obviously satisfied. Let us suppose that Y n ∈ S ∞ and |Z n |⋆W ∈ BMO. Then for all t ∈ [0, T ], under (HQ), Itô formula gives the following equality s 2 ds.
By taking conditional expectation we get for every stopping time τ: Taking the essential supremum with respect to τ in following inequalities, we obtain Thus |Z n+1 | ⋆ W ∈ BMO and we have: Using the induction assumption we obtain The induction is achieved. Now we can use Proposition -1.2 with K = 1 8γ 2 1 − 1 − 32γ 2 ξ 2 L ∞ : since Z n ⋆ W tends to Z ⋆ W in H 2 , we conclude that |Z| ⋆ W 2 BMO 1 8γ 2 1 − 1 − 32γ 2 ξ 2 L ∞ . Finally, we use that Y n tends to Y in S 2 to pass to the limit into (5.1) and to obtain the final upper bound on Y S ∞ .
Proof. [of Proposition -2.1] The proof of the proposition is a direct consequence of Theorem -2.3 together with Proposition -5.1: since the map Proof. To simplify notations in the proof, we skip the superscript M on (Y M , Z M ) and f M . The unique solution (Y, Z) of (1.7) can be constructed with a Picard principle. We consider a sequence (Y n , Z n ) n∈N such that We can easily show that replacing Y n+1 by Y n in the generator does not affect the convergence of the scheme since f is a Lipschitz function. Moreover, applying Itô formula to e Kt Y n+1 t 2 with K large enough, we justify with classical inequalities that for all n ∈ N, Y n+1 ∈ S ∞ , with a bound that depend on M for the moment.
Applying Itô formula to the process e −µt Y n+1 t 2 , we obtain Taking conditional expectation, and using assumption (HMon), we get: With the Young inequality we have the following estimate for all n and s ∈ [0, T ]: and thus Finally we obtain Then Once again with an induction we show easily that for all n ∈ N, |Z n | ⋆ W ∈ BMO, Y n ∈ S ∞ and esssup sup Moreover the inequality (5.2) gives us Letting n to infinity, and with the Proposition -1.2, we finally get esssup sup and so we deduce that The last inequality can be used for each started and stopped process T i Z M ⋆ W ⌋T i+1 : esssup sup We are now in position to prove Proposition -2.2.
Proof. [of Proposition -2.2] The previous Remark -5.1 shows that for all M ∈ R + and all h > 0, the process We just have to apply an adaptation of Theorem -2.3 given by Remark -4.7. § 5.3. Proof of Proposition -2. 3 We consider the diagonal framework introduced in section 2 and subsection 4.3. We assume that the generator satisfies (Hdiag), so the generator f can be written as f = f diag (t, z) + g(t, y, z) where f diag is diagonal with respect to z. If we want to apply Theorem -2.2, we have to obtain a uniform estimate on Z M ⋆ W BMO where (Y M , Z M ) is the unique solution of the Lipschitz localized BSDE (1.7). This is the purpose of the following lemma.
Proposition -5.3. Let us assume that there exist nonnegative constants G d and G such that (i) for all (t, y, z) ∈ [0, T ] × R d × R d×k , f diag (t, z) G d |z| 2 , |g(t, y, z)| G |z| 2 . (ii) Then, Z M ⋆ W ∈ BMO, Y M ∈ S ∞ and we have following estimates: Proof. To simplify notations in the proof, we skip once again the superscript M on Y M , Z M and f M . The unique solution (Y, Z) ∈ S 2 R d × H 2 R d×k of (1.7) can be constructed with a Picard principle slightly different than the one used in the seminal paper of Pardoux and Peng (see [PP90]). We consider a sequence (Y n , Z n ) n∈N defined by Obviously, we can easily show that (Y n , Z n ) n∈N tends to (Y, Z ⋆ W ) in S 2 R d × H 2 R d×k since we are in the Lipschitz framework. We will prove by induction that: for all n ∈ N, Y n ∈ S ∞ , |Z n | ⋆ W ∈ BMO and The result is obvious for n = 0. Let us assume that for a given n ∈ N we have Y n ∈ S ∞ R d , |Z n | ⋆W ∈ BMO Since ϕ ′′ (.)− 2G d ϕ ′ (.) = 1, ϕ 0 and ϕ ′ (x) (2G d ) −1 e 2G d Y n+1,i S ∞ whenever |x| Y n+1,i S ∞ , taking the conditional expectation with respect to F τ , we compute Thus, we get the estimate (5.6) By using same arguments as Darling in [Dar95] we can show that Y M is in G almost surely. Indeed, we know with (5.6) applied to F = F dc , integrating between τ and σ together with (ii), that for all σ τ a.s, Let us consider for all n ∈ N the following sequence of stopping times: σ n = inf u τ F dc (Y M u ) 1 n a.s .
Each σ n is finite almost surely since ξ ∈ G. Continuity of Y M gives for all n ∈ N, F dc Y M σ n 1 n a.s. So we get for all stopping time τ: n a.s. and, consequently, P(Y t ∈ G) = 1 for all t ∈ [0, T ]. Moreover the α-strictly doubly convexity on G gives us And finally, continuity of F dc on G yields Thus, assumption (ii) ensures assumption (BMO,m). Since the terminal value is bounded (in G), Theorem -2.3 together with Remark -4.5 gives the result. § 5.5. Proofs of Theorem -2.5 and Theorem -2.6 Proof.
[of Theorem -2.5 and Theorem -2.6] Uniqueness We start by proving Theorem -2.5. Let us consider two continuous Markovian solutions (v, w) and (ṽ,w) such that v andṽ are bounded. We set (t, x) ∈ [0, T ] × R k and we denote (Y t,x , Z t,x ) (resp. (Ỹ t,x ,Z t,x )) the solution of the BSDE (2.5) associated to (v, w) (resp. (ṽ,w)). The idea of the proof is to compare the two solutions by using the stability result given by Remark 4.3. In order to do that, we must show that Z t,x ⋆ W and Z t,x ⋆ W are ε-sliceable BMO martingales. By Remark 2.6 part (2) in [XŽ16], we know that there exist b 0 ∈ R d and (α n ) ∈ (0, 1] N such that v,ṽ ∈ C (α n ),loc b 0 . Moreover, by following same arguments than in the proof of Theorem 2.9 in [XŽ16] we can show that v,ṽ ∈ C (α n ),loc . Finally, we just have to apply Remark -1.1 to conclude that there exists κ ′ ∈ (0, 1] such that v,ṽ ∈ C κ ′ . Now, let us apply the Itô formula to F(Y t,x ): we consider two stopping times τ and σ such that τ σ a.s and we take the conditional expectation x u )f(u, X t,x u ,Y t,x u , Z t,x u ) du + The map F is a Lipschitz function on the centred Euclidean ball of radius v L ∞ ([0,T ]×R k ) ∨ ṽ L ∞ ([0,T ]×R k ) . Denoting by L its Lipschitz constant, we obtain for all n ∈ N, where we have used in the last inequality a classical estimate for SDEs when b and σ are bounded. For N ∈ N * we set T i = iT N . Then, for all i ∈ {0, ..., N − 1} and stopping time T i τ T i+1 we get , and finally, for N large enough, we have that with 2L y K 2 + 2 √ 2L z KC ′ 2 < 1. Obviously, this estimate is also true forZ t,x which means that Z t,x ⋆W andZ t,x ⋆ W are in Z slic,2 BMO . By using Remark -4.3 we get that v(t, x) =ṽ(t, x) and E T t |w(s, X t,x s ) − w ′ (s, X t,x s )| 2 ds = 0. Since this is true for all (t, x) ∈ [0, T ] × R k and, due to (HX), X 0,x s has positive density on R k for s ∈ (0, T ], w =w a.s. with respect to the Lebesgue measure on [0, T ] × R k . Then, (v, w) and (v ′ , w ′ ) are equal.
which is equivalent to take σ (s, .) = 0 and b(s, .) = 0 for s ∈ [0,t ′ ]. By the same token we can extend (Y t,x u , Z t,x u ) t u T to [0,t]. Then, a standard estimate gives us where C does not depend on (t, x) and (t ′ , x ′ ). To conclude we just have to study the remaining term Y t,x t − Y t ′ ,x ′ t . Thanks to calculations done in the uniqueness part of the proof, we know that there exist some deterministic times 0 = T 0 T 1 ... T N = T such that T i Z t,x ⋆ W ⌋T i+1 ∈ Z 2 BMO and T i Z t ′ ,x ′ ⋆ W ⌋T i+1 ∈ Z 2 BMO for all i ∈ {0, ..., N − 1}. Let us emphasize that times 0 = T 0 T 1 ... T N = T can be chosen independently from Denoting by K ε 1 ,ε 2 the constant K ε 1 ,ε 2 := 1 1 − 2mε 2 1 − ε 2 √ 2C ′ m > 0, we have X i S m K i ε 1 ,ε 2 X 0 L m , and finally we obtain The result follows by setting K m,ε 1 ,ε 2 =