RBSDEs And Applications In Mixed Zero-sum Game, In Reversible Investment And In PDEs

An equation in the form ofis so-called Backward Stochastic Differential Equation (BSDE for short).Linear BSDE was first introduced by Bismut in 1973 when he study the maximumprinciple in stochastic optimal control. General nonlinear BSDE was first introduced byPardoux and Peng in 1990. They showed that there is a unique adapted solution whenthe coefficient was Lipshctz continuous. Independently, Duffie and Epstein introducedstochastic differential utilities in economics, as solutions to a certain type of BSDEs.After that time, this kind of equations has received considerable research attention dueto their nice structure and wide applicability in numbers of different areas, such asmathematical finance, stochastic control, economical management and etc..As we have known, BSDE can be widely used in finance. In a complete market, itis possible to construct a portfolio to copy an expected profit which gainsξat time T.The value of the portfolio is given by a BSDE, and Z corresponds to the strategy. Oneof another application is to give the probabilistic representation of PDEs and generalized the classical Feynman-Kac formula to nonlinear case.In this PhD dissertation, we have mainly studied the existence and uniqueness ofsolutions of backward stochastic differential equations (BSDEs for short) as well as theirapplications in mixed zero-sum differential-integral game, in reversible investments andin probabilistic interpretation of solutions of partial integro-differential equations.In thefollowing, we list the main results of this thesis. Chapter 1: In the first chapter, we recall some preliminary results which arerelated to the thesis, present the motivations of our study and list the main result obtained in the thesis.Chapter 2: In this chapter, we are concerned with the problem of existence anduniqueness of a solution for the BSDE driven by a Brownian motion and an independentPoisson measure with two reflecting obstacles (or barriers) which are right continuouswith left limits (rcll for short) processes. More precisely, we focus on the BSDE withoutthe so-called Mokobodski's condition, on one hand, the filtration is generated by aBrownian motion and an independent Poisson random measure and on the other hand,the jumps of the obstacle processes could be either predictable or inaccessible. Weaim at looking for a quintuple of adapted processes (Y, Z, V, K^±) where K^±are rcllnon-decreasing processes such that for any t≤T:where B is a Brownian motion, (?) is an independent compensated Poisson randommeasure and f(t,ω,y,z,v),ξ, L and U are given.With the help of the powerful notion of local solutions, introduced by Hamadenc &Hassani in [52], we show that when the generator of the BSDE is Lipschitz, the obstacleprocesses and their left limits are completely separated, i.e.,then the BSDE (0.0.7) has a unique solution. The main result of this part is given byThm. 2.3.2:Theorem 2.3.2. (Existence and Uniqueness) Under [H], the BSDE (0.0.7) withjumps and two reflecting discontinuous barriers associated with (f,ξ, L, U) has a uniquesolution, i.e., there exits a unique 5-uple (Y, Z,V,K⁺,K^-) which satisfies the BSDE(0.0.7). In the second part of the chapter, we deal with an application in zero-sum mixedstochastic differential-integral game problem. Assume that two agents (or players) c₁and c₂ intervene on a system with antagonistic advantages. The intervention of theagents have two forms, control and stopping and the dynamics of the system whencontrolled is given by:There is a payoff J(u, r; v,σ) between c₁ and c₂ which is a cost (resp. a reward) for c₁(resp. c₂) whose expression is given byIn order to tackle this problem, we introduce a BSDE with two reflecting barriersof type (0.0.7) with a specific generator, which has a unique solution thanks to theprevious result. We show that this game has a value, i.e., the following relation holdstrue:Moreover, the value of the game is expressed by means of a solution of the BSDE.This work completes and closes this problem of zero-sum stochastic games of diffusionprocesses with jumps. Note that, due to the existence of the predictable jumps of thebarriers L and U, the saddle-point of the game does not exist in general. The mainresult of this part is Thm. 2.4.1:Theorem 2.4.1. We have:i. e. Y₀ is the value of the zero-sum mixed differential game.Chapter 3: We study the problem of existence and uniqueness of a solution forthe BSDE with two reflecting barriers (or obstacles) whose coefficients are not squareintegrable. Roughly speaking, a solution for this equation is a quadruple of adaptedstochastic processes (Y, Z, K⁺, K^-) where the processes K^±are continuous and non- decreasing such that for any t∈[0,T]:where B is a Brownian motion,ξ,f, L and U are the given data of the problem.More precisely, on one hand, we focus on weakening the square integrability of thedataξand (f(t,ω,0,0))_t≤T as well as the barriers, which has been relatively few dealtwith. In this chapter, the square integrability of the data is relaxed, i.e.,ξ, sup(?),sup(?) and (?)dt belong only to L^p for some p∈]1,2[. On the otherside, we consider this problem without Mokobodski's condition. This latter supposesthe existence of a difference of two non-negative supermartingales between the obstaclesand which is very difficult to verify in practice. Instead of that, we suppose that the twobarriers are completely separated i.e.,(?).Using again the notion of so-called local solution and the existence and uniqueness result, as well as the comparison,obtained by Hamadene & Popier [63] in the case of BSDE with one reflecting barrier,we prove the existence and uniqueness of solution of (0.0.8) in using the penalizationmethod. The main result of this chapter is Thm. 3.3.1:Theorem 3.3.1. Assume that (?), there is a unique quadruple ofprocesses (Y, Z, K⁺, K^-) solution of the BSDE(0.0.8) associated with (f,ξ, L, U) .Chapter 4: We consider an optimal switching (or starting and stopping) problemwhich has attracted lots of attentions during the last decades because of their widelyapplications in management science, corporate finance, etc.More precisely, on one hand, we consider the switching problem under Knightianuncertainty, i.e., it is not granted that the future uncertainty is characterized by a singleprobability P but a family of probabilities P^u, u∈U, which are equivalent to P. On theother hand, we study the problem for a risk-averse agent, i.e., we take into account ofthe manager's attitude with respect to risk by means of an exponential utility functionand a parameterθ> 0. When the power station is run under the strategyδand thefuture evolves according to the probability P^u, its yield is given by a quantityAs the absolute value ofθdoes not play a crucial role, therefore we will supposeθ=1. The problem that we consider here follows an article of Hamadene & Zhang [65] onthe same subject where the profit J(δ, u) is of type risk-neutral. In order to tackle theproblem, we introduce a verification theorem by means the following system of reflectedBSDEs,where g₁₂, g₂₁ stand for the sunk cost whence the switching happens and H^* is relatedto the Hamilton function.The main result of this part is Thm. 4.2.1:Theorem 4.2.1. There exist two triplet of processes (Yⁱ Zⁱ, Kⁱ), i = 1, 2 which satisfy(0.0.9). Moreover, we have:Finally, the optimal strategy (δ^*, u^*) is given by (?) and for n = 0,…,andFurther, we consider the following system of variational inequalities with interconnected obstacles:whereΦ_i are defined by Note that this system is the deterministic version of verification theorem for the switching problem under Markovian randomness. We give a probabilistic interpretation of thesolution of (0.0.10), more precisely, we show that, the system (0.0.10) has a unique solution in viscosity sense which can be defined by the solutions of a system of reflectedBSDEs of type (0.0.9).Chapter 5: In the last chapter, we consider a probabilistic interpretation forweak Sobolev solution of a semilinear parabolic partial differential integral equation(PIDE for short) by using a forward backward SDE (FBSDE in short). The method isbased on stochastic flow techniques associated with the forward jump diffusion component. Roughly speaking, we study the variational formulation of the following partialdifferential integral equations of parabolic type:where L is the second order differential-integral operator associated with a jump diffusion which is defined component by component byBy developing a stochastic flow method which has been introduced by Bally andMatoussi in [5] in the study of weak solution of stochastic partial differential equations(SPDEs for short), we prove the probabilistic interpretation of the Sobolev solution of(0.0.11). The main result of this chapter is Thm. 5.3.1:Theorem 5.3.1. Under certain assumptions, there exists a unique Sobolev solution uof the PIDE (0.0.11). Moreover, we have the probabilistic representation of the solution:(?), where (?) is the solution of an appropriate Markovian BSDEand, we have, ds (?) dP (?)ρ(x)dx - a.e.,...