BSDEs With Random Default Time, Anticipated BSDEs And Related Results

Backward stochastic differential equations (BSDEs in short) of the following gen-eral form were introduced, in the linear case, by Bismut [14] in 1973 and considered the general form the first time by Pardoux-Peng [73] in 1990.In the last twenty years, the theory of BSDEs has been studied with great interest (see e.g. [1], [2], [4], [20], [21], [22], [28], [54], [62], [63], [64], [68], [83], [90], [91], etc.). Particularly, the comparison theorem turns out to be one of the achievements of this theory. It is due to Peng [79] and then generalized by Pardoux-Peng [74], El Karoui et al. [39], Hu-Peng [55]. It allows to compare the solutions of two BSDEs whenever we can compare the terminal conditions and the generators. Conversely, we can also compare the generators if we can compare the solutions, see e.g. Briand et al. [19], Coquet et al. [30], Hu-Peng [55], Jiang [59].These results are applied widely to default-free markets. Precisely, BSDE was applied widely in financial mathematics, such as the pricing/hedging problem (see e.g. El Karoui et al. [39], etc.), in the stochastic control and game theory (see e.g. Buckdahn-Li [23], El Karoui et al. [39], El Karoui-Hamadene [36], Hamadene [44], Hamadene-Lepeltier [45] and [46], Hamadene et al. [47], Hamadene et al. [48] and [49], Peng [80], Quenez [85], Peng-Wu [81], etc.), and in the theory of partial differential equations (PDEs in short) (see e.g. Barles et al. [5], Barles-Lesigne [6], Briand [18], Pardoux-Peng [74], Pardoux-Tang [76], Pardoux-Veretennikov [77], Peng [78] and [79], Wu-Yu [88], etc.).In the meantime, people also have a good study of the reflected solutions to BS-DEs, that is, the solution is forced to stay above a given stochastic process which is called the obstacle. More precisely, reflected BSDE, with one barrier introduced by El Karoui et al. [37], with double barrier studied by, e.g., Cvitanic-Karatzas [32] and Hamadene et al. [47], is also a hot topic due to its wide applications to finance, the game or control problems and partial differential equations (see e.g. Bally et al. [3], El Karoui et al. [38], Hamadene-Lepeltier [46], Lepeltier et al. [61], Matoussi [70]). These results were then generalized to the case where the obstacle is discontinuous (see e.g. Hamadene [43], Lepeltier-Xu [65], Peng-Xu [82]), and then by Hamadene-Ouknine [50] (see also Hamadene-Wang [51]) to the discontinuous case where the reflected BSDE is driven by a Brownian motion and an independent Poisson random measure.The objective of this thesis is to enrich and improve the theory of BSDEs. In the following, we list the main results of this thesis.Chapter 1:In this chapter, we present the motivations of our work and list the main problems studied from Chapter 2 to Chapter 4.Chapter 2:In this chapter, we introduce a new type of BSDE in a defaultable set-up, called BSDE with random default time, which is driven by Brownian motion as well as a mutually independent martingale appearing in a defaultable setting. These equations are of the following general form: whereγsds is a G-martingale independent of B,τis the random default time and G is the enlarged filtration.For these equations, we have the following result. It should be mentioned here that the comparison theorem requires more conditions than the existence and uniqueness theorem.Theorem 2.2.2. (Existence and Uniqueness Theorem) Assume that g satis-fies (a2.1) and (a2.2), then for any fixed terminal conditionξ∈L2(GT;Rm), the above BSDE has a unique solutionTheorem 2.2.7. (Comparison Theorem) Let (Y, Z,ζ), (Y, Z,ζ) be the unique solu-tions of the following two 1-dimensional BSDEs with random default time, respectively: whereξ,ξsatisfy the same assumptions as in Theorem 2.2.2, g satisfies (a2.1)-(a2.3), gs∈LG2(0,T;R).If thenBesides, the following holds true (the strict comparison theorem):Then we deal with a particular case where More generally, we introduce the PDE approach to default risk via BSDE with random default time. For this, we haveTheorem 2.4.3. Assume that the function u, defined by satisfies, for any given (t, x)∈[0, T]×Rm,Then u has the following probabilistic representation:Moreover, the following hold: where (Yt,x, Zt,x,ζt,x) is determined uniquely by andAs one application, we deal with an application in zero-sum stochastic differential games in a defaultable setting. Assume that two players J1 and J2 intervene on a system with antagonistic advantages. The dynamics of the controlled system is The cost functional corresponding to u∈U andυ∈V is given by which is a cost (resp. reward) for J1 (resp. J2).The conditional expected remaining cost from timeIn order to tackle this problem, we define the Hamilton function associated with this game problem as following: Assume that Isaacs'condition is fulfilled. Denote by (u*,υ*)(t,x,z,ζ) the saddle point for the function H. WriteThe main result isTheorem 2.5.3. The BSDE with random default time has a unique solution which satisfies where Moreover, the pair (u*,v*) is a saddle point for the game. Chapter 3:In this chapter, we consider the anticipated BSDEs (ABSDEs) of the following form: whereδ(·):[0,T]→R+ andζ(·):[0, T]→R+ are continuous functions satisfying certain conditions.We establish the following comparison theorem for multidimensional ABSDEs with generators independent of the anticipated term of Z and possibly not increasing in the anticipated term of Y:Theorem 3.3.5. The following are equivalent: (i) for all the unique solutions to the following ABSDE: satisfy where c> 0 is a constant.Chapter 4:In this chapter, we study the generalized ABSDE (GABSDE) of the following form: For the 1-dimensional GABSDEs, we give a more general comparison theorem as follows:Theorem 4.2.3. Let (j= 1,2) be the unique solutions to GABSDEs respectively: where j= 1,2, fj satisfies C;Rd). If uous semimartingale and then Yt(1)≥Yt(2),a.e.,a.s..Moreover, we are also concerned with the real-valued reflected GABSDE with one continuous barrier S.Theorem 4.3.3. Assume that (A4.1)-(A4.3) hold, then the reflected GABSDE has a unique solutionFortunately, the above result can be applied to deal with a particular case when the obstacle is of functional form For this, we have Theorem 4.4.5. Assume that (a4.1)-(a4.3) hold and moreover that the obstacle S is a semimartingale of the form where (μt)t∈[0,T] and (σt)t∈[0,T] are progressively measurable processes, with values in R and Rd respectively, satisfyingThen the reflected GABSDE has at least a solution and the pair is uniquely determined.