# Khaled Bahlali's research while affiliated with Université de Toulon and other places

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## Publications (68)

In this paper, we study a class of reflected backward stochastic differential equations (RBSDEs) driven by the compensated random measure associated to a given pure jump Markov process X on a general state space U. The reflection keeps the solution above a given càdlàg process. We prove the uniqueness and existence both by a combination of the Snel...

We consider a system of semilinear partial differential equations (PDEs) with a nonlinearity depending on both the solution and its gradient. The Neumann boundary condition depends on the solution in a nonlinear manner. The uniform ellipticity is not required to the diffusion coefficient. We show that this problem admits a viscosity solution which...

This erratum corrects an error in the proof of Theorem 2 in our paper (Bahlali et al., 2019). The main results of the paper remain true as stated.

We consider a system of semilinear partial differential equations (PDEs) with measurable coefficients and a nonlinear Neumann boundary condition. We then construct a sequence of penalized PDEs, which converges to our initial problem. Since the coefficients we consider may be discontinuous, we use the notion of solution in the [Formula: see text]-vi...

Let X X be the solution of a stochastic differential equation in Euclidean space driven by standard Brownian motion, with measurable drift and Sobolev diffusion coefficient. In our main result we show that when the drift is measurable and the diffusion coefficient belongs to an appropriate Sobolev space, the law of X X satisfies Talagrand’s inequal...

We consider a system of semi-linear partial differential equations with measurable coefficients and a nonlinear Neumann boundary condition. We then construct a sequence of penalized partial differential equations which converges to a solution of our initial problem. The solution we construct is in the L p --viscosity sense, since the coefficients c...

Let X be the solution of the multidimensional stochastic differential equationdX(t) = b(t, X(t)) dt + sigma(t, X(t)) dW(t)\, with X(0)=x where W is a standard Brownian motion. We show that when b is measurable and sigma is in an appropriate Sobolev space, the law of X satisfies a uniform quadratic transportation inequality.

We consider various approximation properties for systems driven by a Mc Kean-Vlasov stochastic differential equations (MVSDEs) with continuous coefficients, for which pathwise uniqueness holds. We prove that the solution of such equations is stable with respect to small perturbation of initial conditions, parameters and driving processes. Moreover,...

We prove that probability laws of a backward stochastic differential equation, satisfy a quadratic transportation cost inequality under the uniform metric. That is, a comparison of the Wasserstein distance from the law of the solution of the equation to any other absolutely continuous measure with finite relative entropy. From this we derive concen...

We consider McKean–Vlasov stochastic differential equations (MVSDEs), which are SDEs where the drift and diffusion coefficients depend not only on the state of the unknown process but also on its probability distribution. This type of SDEs was studied in statistical physics and represents the natural setting for stochastic mean-field games. We will...

We introduce a domination argument which asserts that: if we can dominate theparameters of a quadratic backward stochastic differential equation (QBSDE) with continuousgenerator from above and from below by those of two BSDEs having ordered solutions, thenalso the original QBSDE admits at least one solution. This result is presented in a generalfra...

We consider Mc Kean-Vlasov stochastic differential equations (MVSDEs), which are SDEs where the drift and diffusion coefficients depend not only on the state of the unknown process but also on its probability distribution. This type of SDEs was studied in statistical physics and represents the natural setting for stochastic mean-field games. We wil...

Quadratic backward stochastic differential equations with singularity in the value process appear in several applications, including stochastic control and physics. In this paper, we prove existence and uniqueness of equations with generators (dominated by a function) of the form $|z|^2/y$. In the particular case where the BSDE is Markovian, we obt...

We establish a Krylov-type estimate and an Itô-Krylov change of variable formula for the solutions of one-dimensional quadratic backward stochastic differential equations (QBSDEs) with a measurable generator and an arbitrary terminal datum. This allows us to prove various existence and uniqueness results for some classes of QBSDEs with a square int...

We establish the existence and uniqueness of solutions to one dimensional BSDEs with generator allowing a logarithmic growth (|y|| ln |y|| + |z|�| ln |z||) in the state variables y and z. This is done with an Lp− integrable terminal value, for some p > 2. As byproduct, we obtain the existence of viscosity solutions to PDEs with logarithmic nonlinea...

We establish the existence and uniqueness of solutions to one dimensional BSDEs with generator allowing a logarithmic growth ((Formula presented.)) in the state variables y and z. This is done with an (Formula presented.) integrable terminal value, for some (Formula presented.). As byproduct, we obtain the existence of viscosity solutions to PDEs w...

We consider optimal control problems for systems governed by mean-field stochastic differential equations, where the control enters both the drift and the diffusion coefficient. We study the relaxed model, in which admissible controls are measure-valued processes and the relaxed state process is driven by an orthogonal martingale measure, whose cov...

We a controlled system driven by a coupled forward-backward stochastic differential equation (FBSDE) with a non degenerate diffusion matrix. The cost functional is defined by the solution of the controlled backward stochastic differential equation (BSDE), at the initial time. Our goal is to find an optimal control which minimizes the cost functiona...

We establish the existence of an optimal control for a system driven by a coupled forward–backward stochastic differential equation (FBDSE) whose diffusion coefficient may degenerate (i.e. are not necessary uniformly elliptic). The cost functional is defined as the initial value of the backward component of the solution. We construct a sequence of...

We deal with multidimensional backward doubly stochastic differential equations (BDSDEs) with a superlinear growth generator and a square integrable terminal datum. We introduce new local conditions on the generator and then show that they ensure the existence and uniqueness as well as the stability of solutions. Our work goes beyond the previous r...

We establish a Krylov-type estimate and an Itô–Krylov change of variable formula for the solutions of one-dimensional quadratic backward stochastic differential equations (QBSDEs) with a measurable generator and an arbitrary terminal datum. This allows us to prove various existence and uniqueness results for some classes of QBSDEs with a square int...

We establish an averaging principle for a family of solutions$(X^{\varepsilon}, Y^{\varepsilon})$ $ :=$ $(X^{1,\,\varepsilon},\,X^{2,\,\varepsilon},\, Y^{\varepsilon})$ of a system of SDE-BSDEwith a null recurrent fast component $X^{1,\,\varepsilon}$. Incontrast to the classical periodic case, we can not rely on aninvariant probability and the slow...

The purpose of this paper is to study some properties of solutions to one
dimensional as well as multidimensional stochastic differential equations (SDEs
in short) with super-linear growth conditions on the coefficients. Taking
inspiration from \cite{BEHP, KBahlali, Bahlali}, we introduce a new {\it{local
condition}} which ensures the pathwise uniq...

We deal with backward doubly stochastic differential equations (BDSDEs) with a superlinear growth generator and a square integrable terminal datum. We introduce a new local condition on the generator, then we show that it ensures the existence and uniqueness as well as the stability of solutions. Our work goes beyond the previous results on the mul...

In the first part of this paper, we deal with the u nique solvability of multidimensional backward stochastic differential equations (BSDEs) with a p-integrable terminal condition (p > 1) and a superlinear growth generator. We introduce a new local condition on the generator (assumption (H.4)) and then show that it ensures the existence and uniquen...

In a first step, we establish the existence (and sometimes the uniqueness) of
solutions for a large class of quadratic backward stochastic differential
equations (QBSDEs) with continuous generator and a merely square integrable
terminal condition. Our approach is different from those existing in the
literature. Although we are focused on QBSDEs, ou...

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In this paper we prove an approximation result for the viscosity solution of
a system of semi-linear partial differential equations with continuous
coefficients and nonlinear Neumann boundary condition. The approximation we use
is based on a penalization method and our approach is probabilistic. We prove
the weak uniqueness of the solution for the...

We establish the existence and uniqueness of square integrable solutions for a class of one-dimensional quadratic backward stochastic differential equations (QBSDEs). This is done with a merely square integrable terminal condition, and in some cases with a measurable generator. This shows, in particular, that neither the existence of exponential mo...

We consider a semilinear partial differential equation (PDE) of non-divergence form perturbed by a small parameter. We then study the asymptotic behavior of Sobolev solutions in the case where the coefficients admit limits in C`esaro sense. Neither periodicity nor ergodicity will be needed for the coefficients. In our situation, the limit (or avera...

In this paper, we consider a partial information stochastic control problem where the system is governed by a nonlinear stochastic differential equation driven by Teugels martingales associated with some Lévy process and an independent Brownian motion. We prove optimality necessary conditions in the form of a maximum principle. These conditions tur...

In this paper, we study the existence of an optimal strategy for the
stochastic control of diffusion in general case and a saddle-point for zero-sum
stochastic differential games. The problem is formulated as an extended BSDE
with logarithmic growth in the $z$-variable and terminal value in some $L^p$
space. We also show the existence and uniquenes...

This paper investigates the relationship between the stochastic maximum principle and the dynamic programming principle for singular stochastic control problems. The state of the system under consideration is governed by a stochastic differential equation, with nonlinear coefficients, allowing both classical control and singular control. We show th...

We prove the existence of optimal relaxed controls as well as strict optimal controls for systems governed by non linear forward–backward stochastic differential equations (FBSDEs). Our approach is based on weak convergence techniques for the associated FBSDEs in the Jakubowski S-topology and a suitable Skorokhod representation theorem.

We prove the existence of optimal relaxed controls as well as strict optimal controls for systems governed by non linear forward–backward stochastic differential equations (FBSDEs). Our approach is based on weak convergence techniques for the associated \FBSDEs\ in the Jakubowski S-topology and a suitable Skorokhod representation theorem.

We consider control problems for systems driven by linear backward stochastic differential equations (BSDEs). We prove the existence of strict optimal controls under the convexity of the control domain as well as the cost functional. Our approach is based on strong convergence techniques for the associated linear BSDEs. Moreover, we establish neces...

We study multidimensional backward stochastic differential equations (BSDEs) which cover the logarithmic nonlinearity u log u. More precisely, we establish the existence and uniqueness as well as the stability of p-integrable solutions (p > 1) to multidimensional BSDEs with a p-integrable terminal condition and a super-linear growth generator in th...

We establish the existence and uniqueness as well as the stability of p-integrable solutions to multidimensional backward stochastic differential equations (BSDEs) with super-linear growth coefficient and a p-integrable terminal condition (p>1). The generator could neither be locally monotone in the variable y nor locally Lipschitz in the variable...

We consider control problems for systems governed by a nonlinear forward backward stochastic differential equation (FBSDE). We establish necessary as well as sufficient conditions for near optimality, satisfied by all near optimal controls. These conditions are described by two adjoint processes, corresponding to the forward and backward components...

The present paper studies the stochastic maximum principle in singular optimal control, where the state is governed by a stochastic differential equation with nonsmooth coefficients, allowing both classical control and singular control. The proof of the main result is based on the approximation of the initial problem, by a sequence of control probl...

We study the asymptotic behavior of solutions of semilinear PDEs. Neither periodicity nor ergod-icity will be assumed. On the other hand, we assume that the coefficients have averages in the Cesaro sense. In such a case, the averaged coefficients could be discontinuous. We use a proba-bilistic approach based on weak convergence of the associated ba...

We study the asymptotic behavior of solution of semi-linear PDEs. Neither periodicity nor ergodicity will be assumed. In return, we assume that the coefficients admit a limit in \`{C}esaro sense. In such a case, the averaged coefficients could be discontinuous. We use probabilistic approach based on weak convergence for the associated backward stoc...

We establish a stochastic maximum principle in optimal control of a general class of degenerate diffusion processes with global
Lipschitz coefficients, generalizing the existing results on stochastic control of diffusion processes. We use distributional
derivatives of the coefficients and the Bouleau Hirsh flow property, in order to define the adjo...

We prove a Yamada-Watanabe theorem for forward-backward stochastic differential equations (FBSDEs). As a consequence, we show that weak existence and pathwise uniqueness of the solution imply the existence of a strong solution. We use this result to establish existence and uniqueness of weak solutions for a large class of FBSDEs. We also give a pro...

We consider stochastic differential equations for which pathwise uniqueness holds. By using Skorokhod's selection theorem
we establish various strong stability results under perturbation of the initial conditions, coefficients and driving processes.
Applications to the convergence of successive approximations and to stochastic control of diffusion...

We consider stochastic partial differential equations on $\mathbb{R}^{d}, d\geq 1$, driven by a Gaussian noise white in time and colored in space, for which the pathwise uniqueness holds. By using the Skorokhod representation theorem we establish various strong stability results. Then, we give an application to the convergence of the Picard success...

We deal with backward stochastic differential equations with two reflecting barriers and a continuous coefficient which is, first, linear growth in (y,z) and then quadratic growth with respect to z. In both cases we show the existence of a maximal solution.

We study the existence and uniqueness of Reflected Backward Stochastic Differential Equation (RBSDE for short) with both monotone and locally monotone coefficient and squared integrable terminal data. This is done with a polynomial growth condition on the coefficient. An application to the homogenization of multivalued Partial Differential Equation...

We prove that the convergence of the approximation with time delay, as well as pathwise uniqueness, are generic properties in ordinary differential equations as well as in stochastic differential equations. This is done in the case where the coefficients are neither bounded nor time continuous. The approximation with time delay is used to obtain ex...

We prove that in the sense of Baire category, almost all backward stochastic differential equations (BSDEs) with bounded and continuous coefficient have the properties of existence and uniqueness of solutions as well as the continuous dependence of solutions on the coefficient and the L2-convergence of their associated successive approximations.

We deal with quasi–linear parabolic stochastic partial differential equations. We prove that in the sense of Baire category, almost all quasi–linear parabolic stochastic partial differential equations (SPDE) with continuous coefficient have the properties of existence and uniqueness of solutions, as well as the continuous dependence of solutions on...

We prove an existence and uniqueness result for backward
stochastic differential equations whose coefficients satisfy a
stochastic monotonicity condition. In this setting, we deal with both
constant and random terminal times. In the random case, the
terminal time is allowed to take infinite values.
But in a Markovian framework, that is coupled with...

We deal with reflected backward stochastic differential equations in a d-dimensional convex region with super-linear growth coefficient. We prove, in this setting, various existence and uniqueness results. This is done with an unbounded terminal data.

We deal with backward stochastic differential equations (BSDE for short) driven by
Teugel's martingales and an independent Brownian motion. We study the existence,
uniqueness and comparison of solutions for these equations under a Lipschitz as well as
a locally Lipschitz conditions on the coefficient. In the locally Lipschitz case, we prove
that if...

We prove existence, uniqueness and stability of the solution for multidimensional backward stochastic differential equations (BSDE) with locally monotone coefficient. This is done with an almost quadratic growth coefficient and a square integrable terminal data. The coefficient could be neither locally Lipschitz in the variable y nor in the variabl...

We deal with multidimensional backward stochastic differential equations (BSDE) with locally Lipschitz coefficient in both variables y, z and an only square integrable terminal data. Let L N be the Lipschitz constant of the coefficient on the ball B(0, N) of R d × R dr . We prove that if L N = O(√ log N), then the corresponding BSDE has a unique so...

We prove the existence and uniqueness of the solution of reflected multidimensional backward stochastic differential equation in d-dimensional convex region with locally Lipschitz coefficient and squared integrable terminal condition.

We deal with multidimensional backward stochastic differential equations (BSDE) with locally Lipschitz (in both variables y,z) and sublinear growth coefficient and, an only square integrable terminale data. Let B(0,N) denote the ball of and the Lipschitz constant on B(0,N) of the coefficient. We prove that if , then the corresponding BSDE has a uni...

In this paper, we study stochastic differential equations driven by semi-martingales, with time dependent and non Lipschitz coefficients. The properties established are based on an improvement of Krylov's inequality for semi-martingales. Under a condition of absolute continuity of the quadratic Variation process associated to the martingale part, w...

We prove that in the sense of Baire category, almost all BSDE with bounded continuous coefficients have a unique Solution.

Both Itô's stochastic differential equations, as well as equations driven by semimartingales, with non degenerate diffusion coefficient, are considered. Multidimensional pathwise uniqueness and non-contact property, as well as one dimensional homeomorphic property, of solutions, are studied under weak conditions on the coefficients. It will be show...

In this paper, we study the necessary conditions for optimality of a control problem where the state process is governed by a stochastic differential equation, with non differentiable coefficients. We obtain a stochastic maximum principle for this model. This is the first version of the stochastic maximum principle that covers the non smooth case....

Property P is said to be generic for a class of stochastic differential equations, if P is satisfied by each equation in , where is a set of first category of Baire in It is proved that the existence, the pathwise uniqueness of strong solution, the convergence of successive approximations and the convergence of Euler-scheme are generic properties f...

We establish an estimate, related to the solutions of Itô’s stochastic differential equation, from which we deduce (via exponential semi-martingales) various properties of the flow associated to the solution of multidimensional stochastic differential equation with Sobolev space valued diffusion coefficient and measurable drift. We extend some resu...

We give a solution for multidimensional forward backward stochastic dierential equa- tions when the generator is only continuous and with super-liner growth. We highlight the connection of such FBSDEs with degenerate semi-linear partial dierential equations system in the concept of Sobolev solutions.

We consider multidimensional backward stochastic differential equations (BSDEs). We prove the existence and uniqueness of solutions when the coefficient grow super-linearly, and moreover, can be neither locally Lipschitz in the variable y nor in the variable z. This is done with super-linear growth coefficient and a p-integrable terminal condition...

## Citations

... This allows us to deal with the case where the nonlinearity depends on both the solution and its gradient. Our work extends, in particular, the result of [4] and, in some sense, those of [1,3]. In contrast to works [1, 3, 4], we do not pass by the weak compactness of the laws of the stochastic system associated to our problem. ...

... Regarding Transportations inequalities for backward SDEs, to the best of the authors' knowledge the only work on the subject is the paper [3] available online since August 2019. It uses the Girsanov transform technique of [15] to derive quadratic transportation inequalities for laws of one-dimensional BSDEs with bounded coefficients, and Lipschitz continuous generators. ...

... It is not only because the IPS itself is an important model in molecular dynamics and quantum mechanics, but also because the IPS has a mathematically well-defined mean-limit [5][6][7][8] as the number of particles tends to infinity. The mean-field dynamics of the IPS is a distribution-dependent SDE, also known as the McKean-Vlasov process (MVP), which is an important model in statistical physics describing the ensemble behavior of a system of particles [9,10]. In this paper we focus on the simplest IPS model with only pairwise interactions. ...

... We are interested in the unbounded solution of BSDE (1.1) with a super-linearly growing generator g, which satisfies, roughly speaking, that dP × dt-a.e., ∀ (y, z) ∈ R × R d , |g(ω, t, y, z)| ≤ α t (ω) + β|y|(ln |y|) δ 1 |y|>1 + γ|z|| ln |z|| λ (1. 2) for some (δ, λ) ∈ (0, 1] × [0, +∞). ...

... Finally, we would like to especially mention that Bahlali et al. [3,7] proved the existence of a solution to BSDE (ξ, g) in the space S p × M 2 for some sufficiently large p > 2, when the terminal condition (ξ, α · ) belongs to L p × L p and the generator g satisfies (1.2) with δ = 1 and λ = 1/2. They also established the uniqueness of the solution when g further satisfies a locally monotonicity condition in (y, z). ...

... The theory and the stochastic calculus for G-SDE have been developed by Peng and co-workers [39,12]. Relevant preliminary work on existence and uniqueness of fully coupled FBSDE, G-FBSDE and the corresponding dynamic programming (Hamilton-Jacobi Bellman or HJB) equations is due to Redjil & Choutri [42] and Kebiri et al. [4,3,2,26], showing the existence of a relaxed control based on results of El-Karoui et al. [25]. ...

... Their approach was followed extensively by El Karoui et al. (1987), Haussmann and Lepeltier (1990), Kurtz and Stockbridge (1998), Mezerdi and Bahlali (2002) and Dufour and Stockbridge (2012). Relaxed controls have also been used to study singular control problems (Haussmann and Suo, 1995;Kurtz et al., 2001;Andersson, 2009), mean-field games (Lacker, 2015;Fu and Horst, 2017;Cecchin and Fischer, 2020;Benazzoli et al., 2020;Bouveret et al., 2020;Barrasso and Touzi, 2020), mean-field control problems (Bahlali et al., 2017), continuous-time reinforcement learning , and optimal control of piece-wise deterministic Markov processes (Costa and Dufour, 2010a,b;do Valle Costa and Dufour, 2013;Bäuerle and Rieder, 2009;Bauerle and Lange, 2018). ...

... Since g is bounded, the results in Kobylanski 30 entail existence and uniqueness of the FBSDE (20); cf. Bahlali et al. 1 . As a consequence, Y 0 = V (x) equals the value function of our control problem. ...

... A number of studies on BDSDEs follow the work of Pardoux and Peng, see e.g. [5,10,25,31,32,37]. In the setting of CONTACT Roger Pettersson roger.pettersson@lnu.se ...

... For BSDEs with quadratic or sub-quadratic growth in z, Briand and Hu [10], Bahlali [5], Geiss and Ylinen [19] (all in case of BSDEs driven by a Brownian motion) and Antonelli and Mancini [3] (for BSDEs with jumps and finite Lévy measure), investigate the requirements on the terminal condition such that existence and uniqueness of solutions holds. It is well-known that -in the case of quadratic growth in z, -square integrability of ξ is not sufficient but the assumption that ξ is bounded can be relaxed. ...