MATOUSSI Anis

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The aim of this short note is to fill in a gap in our earlier paper [7] on 2BSDEs with reflections, and to explain how to correct the subsequent results in the second paper [6]. We also provide more insight on the properties of 2RBSDEs, in the light of the recent contributions [5, 13] in the so-called G−framework.

We consider a stylized model for a power network with distributed local power generation and storage. This system is modeled as network connection a large number of nodes, where each node is characterized by a local electricity consumption, has a local electricity production (e.g. photovoltaic panels), and manages a local storage device. Depending on its instantaneous consumption and production rates as well as its storage management decision, each node may either buy or sell electricity, impacting the electricity spot price. The objective at each node is to minimize energy and storage costs by optimally controlling the storage device. In a non-cooperative game setting, we are led to the analysis of a non-zero sum stochastic game with $N$ players where the interaction takes place through the spot price mechanism. For an infinite number of agents, our model corresponds to an Extended Mean-Field Game (EMFG). In a linear quadratic setting, we obtain and explicit solution to the EMFG, we show that it provides an approximate Nash-equilibrium for $N$-player game, and we compare this solution to the optimal strategy of a central planner.

This thesis is about Stochastic Retrograde Differential Equations (SRDEs) reflected with two obstacles and their applications to zero-sum switching games, systems of partial differential equations, mean-field problems. There are two parts to this thesis. The first part deals with stochastic optimal switching and is composed of two works. In the first work, we show the existence of the solution of a system of reflexive EDSRs with interconnected bilateral obstacles in the general probabilistic framework. This problem is related to a zero-sum switching game. Then we address the question of the uniqueness of the solution. And finally we apply the obtained results to show that the associated PDE system has a unique solution in the viscosity sense, without the usual monotonicity condition. In the second work, we also consider a system of reflected PDEs with interconnected bilateral obstacles in the Markovian framework. The difference with the first work lies in the fact that the switching does not take place in the same way. This time when the switching is done, the system is put in the next state no matter which of the players decides to switch. This difference is fundamental and complicates the problem of the existence of the solution of the system. Nevertheless, in the Markovian framework we show this existence and give a uniqueness result using mainly Perron's method. Then, the link with a specific switching set is established in two settings. In the second part we study the one-dimensional reflected two-obstacle mean-field EDSR. Using the fixed point method, we show the existence and uniqueness of the solution in two frames, depending on the integrability of the data.

In this paper, we study a doubly Reflected Backward Stochastic Differential Equation with Jumps (DRBSDEs in short) when the driver have general quadratic growth. We extend the result of Essaky and Hassani [14] to the jump setting and a generator with general exponential quadratic growth.

We prove an existence and uniqueness result for two-obstacle problem for quasilinear Stochastic PDEs (DOSPDEs for short). The method is based on the probabilistic interpretation of the solution by using the backward doubly stochastic differential equations (BDSDEs for short).

This work concerns the study of consistent dynamic utilities in a financial market with jumps. We extend the results established in the paper [EKM13] to this framework. The ideas are similar but the difficulties are different due to the presence of the Lévy process. An additional complexity is clearly the interpretation of the terms of jumps in the different problems primal and dual one and relate them to each other. To do, we need an extension of the Itô-Ventzel's formula to jump's frame. By verification, we show that the dynamic utility is solution of a non-linear second order stochastic partial integro-differential equation (SPIDE). The main difficulty is that this SPIDE is forward in time, so there are no results in the literature that ensure the existence of a solution or simply allow us to deduce important properties, in our study, such as concavity or monotonicity. Our approach is based on a complete study of the primal and the dual problems. This allows us, firstly, to establish a connection between the utility-SPIDE and two SDEs satisfied by the optimal processes. Based on this connection and the SDE's theory, stochastic flow technics and characteristic method allow us, secondly, to completely solve the equation.

This thesis deals with the study of backward stochastic differential equations (BSDEs) with jumps and their applications.In Chapter 1, we study a class of BSDEs when the noise comes from a Brownian motion and an independent random jump measure with infinite activity. More precisely, we treat the case where the generator is quadratically increasing and the terminal condition is unbounded. The existence and uniqueness of the solution are proved by combining both the monotonic approximation procedure and a stepwise approach. This method allows to solve the case where the terminal condition is unbounded.Chapter 2 is devoted to generalized doubly reflected jumping RLS under weak integrability assumptions. More precisely, we show the existence of a solution for a stochastic quadratically growing generator and an unbounded terminal condition. We also show, in an appropriate framework, the connection between our class of backward stochastic differential equations and zero-sum games.In chapter 3, we consider a general class of coupled progressive-retrograde RDEs with Mackean Vlasov type jumps under a weak monotonicity condition. Existence and uniqueness results are established under two classes of assumptions based on perturbation schemes of either the progressive stochastic differential equation or the retrograde stochastic differential equation. The chapter is concluded with a problem of optimal energy storage in a medium field electric park.

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In this paper, we prove the existence and uniqueness of the solution of a coupled Mean-Field Forward-Backward SDE system with Jumps. Then, we give an application in the field of storage problem in smart grids, studied in [4] in the case where the production of electricity is not predictable due, for example, to the changes in meteorological forecasts.

This thesis is devoted to the study of stochastic backward differential equations (SDEs) and their applications. In Chapter 1, we study the terminal wealth utility maximization problem where the asset price can be discontinuous under constraints on the agent's strategies. We focus on the SRDE whose solution represents the maximum utility, which allows us to transfer results on quadratic SRDEs, in particular stability results, to the utility maximization problem. In Chapter 2, we consider the American option pricing problem from both theoretical and numerical perspectives based on the representation of the option price as a viscosity solution of a nonlinear parabolic equation. We extend the result proved in [Benth, Karlsen and Reikvam 2003] for an American put or call to a more general case in a multidimensional framework. We propose two numerical schemes inspired by branching processes. Our numerical experiments show that the approximation of the discontinuous generator, associated to the PDE, by local polynomials is not efficient while a simple randomization procedure gives very good results. In chapter 3, we prove existence and uniqueness results for a general class of mean-field progressive-retrograde equations under a weak monotonicity condition and a non-degeneracy assumption on the progressive equation and we give an application in the field of energy storage in the case of unpredictable electricity production.

We consider a stylized model for a power network with distributed local power generation and storage. This system is modeled as network connection a large number of nodes, where each node is characterized by a local electricity consumption, has a local electricity production (e.g. photovoltaic panels), and manages a local storage device. Depending on its instantaneous consumption and production rates as well as its storage management decision, each node may either buy or sell electricity, impacting the electricity spot price. The objective at each node is to minimize energy and storage costs by optimally controlling the storage device. In a non-cooperative game setting, we are led to the analysis of a non-zero sum stochastic game with $N$ players where the interaction takes place through the spot price mechanism. For an infinite number of agents, our model corresponds to an Extended Mean-Field Game (EMFG). In a linear quadratic setting, we obtain and explicit solution to the EMFG, we show that it provides an approximate Nash-equilibrium for $N$-player game, and we compare this solution to the optimal strategy of a central planner.

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This paper introduces a dual problem to study a continuous-time consumption and investment problem with incomplete markets and stochastic differential utility. For Epstein-Zin utility, duality between the primal and dual problems is established. Consequently the optimal strategy of the consumption and investment problem is identified without assuming several technical conditions on market model, utility specification, and agent's admissible strategy. Meanwhile the minimizer of the dual problem is identified as the utility gradient of the primal value and is economically interpreted as the "least favorable" completion of the market.

In this paper, we study a class of Quadratic Backward Stochastic Differential Equations (QBSDE in short) with jumps and unbounded terminal condition. We extend the class of quadratic semimartingales introduced by Barrieu and El Karoui (2013) in the jump diffusion model. The properties of these class of semimartingales lead us to prove existence result for the solution of a quadratic BSDEs.

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A parametric framework is proposed to model both attritional and atypical claims for insurance pricing. This model relies on a classical Generalized Linear Model for attritional claims and a non-standard Generalized Pareto distribution regression model for atypical claims. Maximum likelihood estimators (closed-form for the Generalized Linear Model part and computed with Iterated Weighted Least Square procedure for the Generalized Pareto distribution regression part) are proposed to calibrate the model. Two premium principles (expected value principle and standard deviation principle) are computed on a real data set of fire warranty of a corporate line-of-business. In our methodology, the tuning of the safety loading in the two premium principles is performed to meet a solvency constraint so that the premium caps a high-level quantile of the aggregate annual claim distribution over a reference portfolio.

In this paper, we present a sufficient condition for the large deviation criteria of Budhiraja, Dupuis and Maroulas for functionals of Brownian motions. We then establish a large deviation principle for obstacle problems of quasi-linear stochastic partial differential equations. It turns out that the backward stochastic differential equations will play an important role.

In this article, we are interested in solving numerically backward doubly stochastic differential equations (BDSDEs) with random terminal time tau. The main motivations are giving a probabilistic representation of the Sobolev's solution of Dirichlet problem for semilinear SPDEs and providing the numerical scheme for such SPDEs. Thus, we study the strong approximation of this class of BDSDEs when tau is the first exit time of a forward SDE from a cylindrical domain. Euler schemes and bounds for the discrete-time approximation error are provided.

In this paper, a stochastic control problem under model uncertainty with general penalty term is studied. Two types of penalties are considered. The first one is of type f-divergence penalty treated in the general framework of a continuous filtration. The second one called consistent time penalty studied in the context of a Brownian filtration. In the case of consistent time penalty, we characterize the value process of our stochastic control problem as the unique solution of a class of quadratic backward stochastic differential equation with unbounded terminal condition.

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This paper presents existence and uniqueness results for reflected backward doubly stochastic differential equations (in short RBDSDEs) in a convex domain D without any regularity conditions on the boundary. Moreover, using a stochastic flow approach a probabilistic interpretation for a system of reflected SPDEs in a domain is given via such RBDSDEs. The solution is expressed as a pair (u, ν) where u is a predictable continuous process which takes values in a Sobolev space and ν is a random regular measure. The bounded variation process K, the component of the solution of the reflected BDSDE, controls the set when u reaches the boundary of D. This bounded variation process determines the measure ν from a particular relation by using the inverse of the flow associated to the the diffusion operator.

We prove an L2-regularity result for the solutions of Forward Backward Doubly Stochastic Differentiel Equations (F-BDSDEs in short) under globally Lipschitz continuous assumptions on the coefficients. Therefore, we extend the well known regularity results established by Zhang (2004) for Forward Backward Stochastic Differential Equations (F-BSDEs in short) to the doubly stochastic framework. To this end, we prove (by Malliavin calculus) a representation result for the martingale component of the solution of the F-BDSDE under the assumption that the coefficients are continuous in time and continuously differentiable in space with bounded partial derivatives. As an (important) application of our L2-regularity result, we derive the rate of convergence in time for the (Euler time discretization based) numerical scheme for F-BDSDEs proposed by Bachouch et al.(2016) under only globally Lipschitz continuous assumptions.

The aim of this short note is to fill in a gap in our earlier paper [7] on 2BSDEs with reflections, and to explain how to correct the subsequent results in the second paper [6]. We also provide more insight on the properties of 2RBSDEs, in the light of the recent contributions [5, 13] in the so-called G−framework.

This thesis deals with second order reflected stochastic backward differential equations in a general filtration . We have first treated the reflection at a lower barrier and then extended the result to the case of an upper barrier. Our contribution consists in proving the existence and uniqueness of the solution of these equations in the framework of a general filtration under weak assumptions. We replace the uniform regularity by the Borel type regularity. The dynamic programming principle for the robust stochastic control problem is thus proved under weak assumptions, i.e. without regularity on the generator, the terminal condition and the barrier. In the framework of standard Stochastic Retrograde Differential Equations (SRDEs), the upper and lower barrier reflection problems are symmetric. However, in the framework of second-order SRDEs, this symmetry is no longer valid because of the nonlinearity of the expectation under which our non-dominated robust stochastic control problem is defined. Then we present a numerical approximation scheme for a class of reflected second order SDEs. In particular we show the convergence of the scheme and we numerically test the obtained results.

In this article, we propose a wellposedness theory for a class of second order backward doubly stochastic differential equation (2BDSDE). We prove existence and uniqueness of the solution under a Lipschitz type assumption on the generator, and we investigate the links between our 2BDSDEs and a class of parabolic fully nonLinear Stochastic PDEs. Precisely, we show that the Markovian solution of 2BDSDEs provide a probabilistic interpretation of the classical and stochastic viscosity solution of fully nonlinear SPDEs.

This thesis studies risk management and hedging using Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR) as risk measures. The first part proposes a price evolution model that we confront with real data from the Paris stock exchange (Euronext PARIS). Our model takes into account the probabilities of occurrence of extreme losses and the regime changes observed in the data. Our approach consists in detecting the different periods of each regime by constructing a hidden Markov chain and estimating the tail of each regime distribution by power laws. We show empirically that the latter are more suitable than normal and stable distributions. The VaR estimation is validated by several backtests and compared to the results of other classical models on a base of 56 stock assets. In the second part, we assume that stock prices are modeled by exponential Lévy processes. First, we develop a numerical method for computing the cumulative VaR and CVaR. This problem is solved using the formalization of Rockafellar and Uryasev, which we evaluate numerically by Fourier inversion. In a second step, we focus on minimizing the hedging risk of European options, under a budget constraint on the initial capital. By measuring this risk by the CVaR, we establish an equivalence between this problem and a Neyman-Pearson type problem, for which we propose a numerical approximation based on the relaxation of the constraint.

In this article, we are interested in solving numerically backward doubly stochastic differential equations (BDSDEs) with random terminal time tau. The main motivations are giving a probabilistic representation of the Sobolev's solution of Dirichlet problem for semilinear SPDEs and providing the numerical scheme for such SPDEs. Thus, we study the strong approximation of this class of BDSDEs when tau is the first exit time of a forward SDE from a cylindrical domain. Euler schemes and bounds for the discrete-time approximation error are provided.

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This thesis is devoted to the study of some problems in the field of stochastic backward differential equations (SDEs), and their applications to partial differential equations.In the first chapter, we introduce the notion of doubly stochastic backward differential equation (DSDE) with singular terminal condition. We first study the EDDSR with monotone generator, and then obtain an existence result by an approximation scheme. A last section establishes the link with stochastic partial differential equations, via the weak solution approach developed by Bally, Matoussi in 2001.The second chapter is devoted to EDDSR with singular terminal condition and jumps. As in the previous chapter, the tricky part will be to prove the continuity in T. We formulate sufficient conditions on the jumps to obtain the latter. A section then establishes the link between minimal solution of the EDSR and integro-differential equations. Finally the last chapter is dedicated to doubly reflected second order stochastic backward differential equations (2EDSR). We have established the existence and uniqueness of such equations. Thus, we first focused on the top barrier reflection problem of 2EDSR. We then combined these results with the existing ones in order to give a correct framework to the 2EDSRDR. The uniqueness is a consequence of a representation property and the existence is obtained by using shift spaces, and regular conditional probability distributions. Finally an application to Dynkin games and Israeli options is discussed in the last section.

In this paper, we first prove existence and uniqueness of the solution of a backward doubly stochastic differential equation (BDSDE) and of the related stochastic partial differential equation (SPDE) under monotonicity assumption on the generator. Then we study the case where the terminal data is singular, in the sense that it can be equal to +∞ on a set of positive measure. In this setting we show that there exists a minimal solution, both for the BDSDE and for the SPDE. Note that solution of the SPDE means weak solution in the Sobolev sense.

In this paper we obtain a Wong-Zakai approximation to solutions of backward doubly stochastic differential equations.

In this paper we design a numerical scheme for approximating Backward Doubly Stochastic Differential Equations (BDSDEs for short) which represent solution to Stochastic Partial Differential Equations (SPDEs). We first use a time-discretization and then, we decompose the value function on a functions basis. The functions are deterministic and depend only on time-space variables, while decomposition coefficients depend on the external Brownian motion B. The coefficients are evaluated through a empirical regression scheme, which is performed conditionally to B. We establish non asymptotic error estimates, conditionally to B, and deduce how to tune parameters to obtain a convergence conditionally and unconditionally to B. We provide numerical experiments as well.

The objective of this thesis is the study of the probabilistic representation of different classes of nonlinear EDPSs (semi-linear, completely nonlinear, reflected in a domain) using the double stochastic backward differential equations (EDDSRs). This thesis contains four different parts. In the first part, we treat the second order EDDSRs (2EDDSRs). We show the existence and uniqueness of solutions of EDDSRs using quasi-secure stochastic control techniques. The main motivation of this study is the probabilistic representation of completely nonlinear EDPSs. In the second part, we study weak Sobolev-type solutions of the obstacle problem for integro-differential partial differential equations (IDDEs). More precisely, we show the Feynman-Kac formula for the EDPIDs via the reflected backward stochastic differential equations with jumps (EDSRRs). More precisely, we establish the existence and uniqueness of the solution of the obstacle problem, which is considered as a pair consisting of the solution and the reflection measure. The approach used is based on the stochastic flow techniques developed in Bally and Matoussi (2001) but the proof is much more technical. In the third part, we treat the existence and uniqueness for EDDSRRs in a convex domain D without any regularity condition on the boundary. Moreover, using the stochastic flow techniques approach we prove the probabilistic interpretation of the weak Sobolev-type solution of a class of reflected EDPSs in a convex domain via EDDSRRs. Finally, we focus on the numerical resolution of random terminal time EDDSRs. The main motivation is to give a probabilistic representation of the Sobolev solutions of semi-linear EDPSs with zero Dirichlet condition at the edge. In this section, we study the strong approximation of this class of EDDSRs when the random terminal time is the first exit time of a cylindrical domain EDS. Thus, we give bounds for the approximation error in discrete time. This section concludes with numerical tests that demonstrate that this approach is effective.

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We give a probabilistic interpretation for the weak Sobolev solution of obstacle problem for semilinear parabolic partial integro-differential equations (PIDE). The results of Léandre [29] about the homeomorphic property for the solution of SDE with jumps are used to construct random test functions for the variational equation for such PIDE. This yields to the natural connection with the associated Reflected Backward Stochastic Differential Equations with jumps (RBSDE), namely the Feynman Kac's formula for the solution of the PIDE. MSC: 60H15. 60G46. 35R60 Keyword: Reflected backward stochastic differential equation, partial parabolic integro-differential equation, jump diffusion process, obstacle problem, stochastic flow, flow of diffeo-morphism.

The aim of this paper is twofold. First, we extend the results of Matoussi et al. (2013) concerning the existence and uniqueness of second-order reflected 2BSDEs to the case of two obstacles. Under some regularity assumptions on one of the barriers, similar to the ones in Crépey and Matoussi (2008), and when the two barriers are completely separated, we provide a complete wellposedness theory for doubly reflected second-order BSDEs. We also show that these objects are related to non-standard optimal stopping games, thus generalizing the connection between DRBSDEs and Dynkin games first proved by Cvitanić and Karatzas (1996). More precisely, we show under a technical assumption that the second order DRBSDEs provide solutions of what we call uncertain Dynkin games and that they also allow us to obtain super and subhedging prices for American game options (also called Israeli options) in financial markets with volatility uncertainty.

The objective of this thesis is the study of a numerical scheme for the approximation of solutions of doubly stochastic backward differential equations (DSDEs). During the last two decades, several methods have been proposed to allow the numerical solution of standard backward stochastic differential equations. In this thesis, we propose an extension of one of these methods to the doubly stochastic case. Our numerical method allows us to attack a wide range of nonlinear stochastic partial differential equations (SPDEs). This is possible through their probabilistic representation in terms of EDDSRs. In the last part, we study a new particle method in the context of neutron protection studies.

The problem of robust utility maximization in an incomplete market with volatility uncertainty is considered, in the sense that the volatility of the market is only assumed to lie between two given bounds. The set of all possible models (probability measures) considered here is non-dominated. We propose to study this problem in the framework of second-order backward stochastic differential equations (2BSDEs for short) with quadratic growth generators. We show for exponential, power and logarithmic utilities that the value function of the problem can be written as the initial value of a particular 2BSDE and prove existence of an optimal strategy. Finally several examples which shed more light on the problem and its links with the classical utility maximization one are provided. In particular, we show that in some cases, the upper bound of the volatility interval plays a central role, exactly as in the option pricing problem with uncertain volatility models of [2].

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In this article, we build upon the work of Soner, Touzi and Zhang [Probab. Theory Related Fields 153 (2012) 149-190] to define a notion of a second order backward stochastic differential equation reflected on a lower c\'{a}dl\'{a}g obstacle. We prove existence and uniqueness of the solution under a Lipschitz-type assumption on the generator, and we investigate some links between our reflected 2BSDEs and nonclassical optimal stopping problems. Finally, we show that reflected 2BSDEs provide a super-hedging price for American options in a market with volatility uncertainty.

In the first part of this thesis, we obtain the existence of a density and Gaussian estimates for the solution of a backward stochastic differential equation. This is an application of Malliavin's calculus and more particularly of a formula of I. Nourdin and F. Viens. The second part of this thesis is devoted to the simulation of a stochastic partial differential equation by a probabilistic method based on the representation of the stochastic partial differential equation in terms of a backward stochastic differential equation, introduced by E. Pardoux and S. Peng. We extend in this framework the ideas of F. Zhang and E. Gobet et al. on the simulation of a backward stochastic differential equation. In the last part, we study the weak error of the implicit Euler scheme for diffusion processes and the stochastic heat equation. In the first case, we extend the results of D. Talay and L. Tubaro. In the second case, we extend the work of A. Debussche.

The main objective of this thesis is to study some financial mathematics problems in an incomplete market with model uncertainty. Recently, the theory of second order backward stochastic differential equations (2EDSRs) has been developed by Soner, Touzi and Zhang on this topic. In this thesis, we adopt their viewpoint. This thesis contains four parts in the area of 2EDSRs. We start by generalizing the theory of 2EDSRs initially introduced in the case of continuous lipschitzian generators to the case of quadratically growing generators. This new class of 2EDSRs will then allow us to study the robust utility maximization problem in non-dominated models. In the second part, we study this problem for three utility functions. In each case, we give a characterization of the value function and of an optimal investment strategy via the solution of a 2EDSR. In the third part, we also provide an existence and uniqueness theory for second-order reflected EDSRs with lower obstacles and lipschitzian generators. We then apply this result to the study of the American option pricing problem in a financial model with uncertain volatility. In the fourth part, we study 2EDSRs with jumps. In particular, we prove the existence of a unique solution in an appropriate space. As an application of these results, we study a robust exponential utility maximization problem with model uncertainty. The uncertainty affects both the volatility process and the measurement of jumps.

This thesis deals with Quasilinear Stochastic Partial Differential Equations. It is divided into two parts. The first part deals with the obstacle problem for quasilinear stochastic partial differential equations and the second part is devoted to the study of quasilinear stochastic partial differential equations driven by a Brownian G-movement. In the first part, we first show the existence and uniqueness of an obstacle problem for quasilinear stochastic partial differential equations (in short OSPDE). Our method is based on analytical techniques coming from the parabolic potential theory. The solution is expressed as a pair (u,v) where u is a continuous predictable process that takes its values in a Sobolev space and v is a random regular measure satisfying the Skohorod condition. Then, we establish a maximum principle for the local solution of quasilinear stochastic partial differential equations with obstacle. The proof is based on a version of Itô's formula and estimates for the positive part of a local solution which is negative on the edge of the considered domain. The objective of the second part is to study the existence and uniqueness of the solution of stochastic partial differential equations directed by G-Brownian motion in the framework of a space with sublinear expectation. We establish an Itô formula for the solution and a comparison theorem.

This thesis deals with the optimization of asset portfolios subject to default risk. The current crisis has allowed us to understand that it is important to take into account the risk of default to be able to give the real value of its portfolio. Indeed, due to the different exchanges of the financial market actors, the financial system has become a network of several connections which it is essential to identify in order to evaluate the risk of investing in a financial asset. In this thesis, we define a financial system with a finite number of connections and we propose a model of the dynamics of an asset in such a system by taking into account the connections between the different assets. The measurement of the correlation will be done through the jump intensity of the processes. Using Stochastic Differential Backward Equations (SDGE), we will derive the price of a contingent asset and take into account the model risk in order to better evaluate the optimal consumption and wealth if one invests in such a market.

The purpose of this thesis is to study the existence and uniqueness of solutions of reflected backward stochastic differential equations and then to link this notion to problems such as the reversible investment or stop-and-go problem, the stochastic differential zero-sum game (mixed type or Dynkin type), or the probabilistic interpretation of weak solutions of partial differential equations in the viscosity sense or in the Sobolev sense in the different settings

The purpose of this thesis is, on the one hand, to study reflected stochastic backward differential equations (SRDEs) and, on the other hand, to prove the existence and uniqueness of solutions of quasi-linear stochastic partial differential equations (SPDEs), formulated in a weak sense . using generalized solutions of doubly stochastic backward differential equations (DSDEs). In the first part, we try to show the existence of a solution for the EDDSR reflected on one or two barriers with non Lipschitz coefficient. We question the minimal assumptions to be included to obtain this result. In the second part, we are interested in the following quasi-linear EDPS: U/T = LU (T, X) + F(T, X, U(T, X), (*U)(T, X))DT + H(T, X, U(T, X), (*U)(T, X))B/T(T), U(T, X) = G(X) or G is a distribution. Given the results already known on this subject, we answer the following questions: - In the case where the coefficients F(S, X, Y, Z) and H(S, X, Y, Z) are linear in (Y, Z) and belong to a Sobolev-like space in X, is there a weak EDDSR formulation to give a Feynman-Kac formula for the EDPS solution? - in the case where the coefficients are nonlinear, can we show the existence and uniqueness of a solution of EDPS and thus generalize the results obtained by Barles and Lesigne (1997) in the framework of standard PDEs?