Reflected Backward Stochastic Differential Equation And Backward Stochastic Volterra Integral Equation Driven By Levy Process

Nonlinear backward stochastic differential equation (BSDE in short) was first in-troduced by Pardoux and Peng in 1990. In 1992, Duffie and Epstein [29] introduced stochastic differential utilities in economics, as solutions to a certain type of BSDEs. More precisely, a classical BSDE defined on [0, T] having the following type: where, the function f(t, y,z) is called the coefficient of BSDE (1.1), and the couple (T,?) is the terminal condition for BSDE (1.1). In Pardoux and Peng [75], the authors proved the existence and uniqueness of adapted solution to BSDE (1.1) providing that the coefficient f(s,y,z) is Lipschitz in both variables y and z, and that?and (f(t,0,0))0?t?T are square integrable. In the past two decades, there are plenty of efforts devoted to the study of the theory of BSDE. At the same time, this fundamental theory has a widely applications, especially in the fields of mathematics finance, stochastic optimal control, nonlinear expectation and PDE, etc (see El Karoui et al. [33], El Karoui ct al. [34], Peng [82] and the cite therein).Later in 1997, El Karoui, Kapoudjian, Pardoux, Peng and Qucncz [32] introduced the concept of reflected BSDE (RBSDE in short), which is actually a backward equation but the solution is forced to stay above a given barrier. To do so, they introduced an increasing process Kt to the original BSDE to push the solution upward. The solution of RBSDE is forced to stay above a given stochastic process, which is called the obstacle. Furthermore, the push is minimal. More precisely, the equation is in the form of This type of BSDEs is motivated by pricing American options and studying the mixed game problems (sec e.g. El Karoui et al. [33], Cvitanic & Karatzas [27], Hamadene & Lcpclticr [42]).Ren et al. [90] established the existence and uniqueness of solutions for the BDS-DEs driven by a Brownian motion and the martingales of Teugels associated with an independent Levy process (BDSDELs) under Lipschitz condition, they also gave a prob-abilistic representation for solutions of stochastic partial differential integral equations.The existence and uniqueness of adapted solution of the previous works arc proved under the Lipschitz assumption on coefficienet. However, the Lipschitz condition is too restrictive to be assumed in many applications.In Nualart & Schoutcns [68], the authors gave a martingale representation theorem associated to Teugel's martingales corresponding to a Levy process; Furthermore, Nu-alart & Schoutens [69] studied the corresponding BSDEs associated to a Levy process. The results were important from a pure mathematical point of view as well as in the world of finance. It could be used for the purpose of option pricing in a Levy market.In 1994, in order to give a probabilistic representation for the solutions of a class of quasilinear stochastic partial differential equations. Pardoux & Peng [77] first con-sidered a class of backward doubly stochastic differential equations (BDSDEs) with two different directions of stochastic integrals. Bahalai et al. [8] obtained the existence and uniqueness of solution for BDSDEs with one continuous lower barrier. However in most of the previous works, solutions arc taken in L2 space or in Lp,p>2. This limits the scope for several applications.In 1991, Hu & Peng [47] considered backward semilinear stochastic evolution equa-tions with values in a complete separable Hilbert space. As a continuation of the pre-vious work, Lin [59] studied a type of backward stochastic Volterra equation under the Lipschitz condition on the coefficient. Later, Yong [101] extended the above equations to a generalized form.. Recently, Ren [87] established the well-posedness of adapted M-solutions for BSVIEs driven by Brownian motion and Poisson random measure. More recently, Wang & Shi [96] first introduced the concept of symmetrical solution (S-solution) to BSVIEs. In this thesis, we are devote to the study of some topics related to the BSDE theory. We establish the existence and uniqueness of adapted solution to RBSDE and RBSDE driven by a Levy process with stochastic Lipschitz coefficient. Then We derive the existence and uniqueness of Lp-solution for reflected backward doubly stochastic differential equations with one continuous lower barrier, p?(1,2). Moreover, we discuss the M-solution and S-solution to backward stochastic Volterra integral equation and backward doubly stochastic Volterra integral equation driven by a Levy process, respectively.In the following, we list the frame and the main results of this thesis.Chapter 1:In this Chapter, we study the reflected BSDE with stochastic Lips-chitz coefficient, the cxistenve and uniqueness of adapted solution is established. More-over, we also study a class of reflected BSDE driven by a Levy process with stochastic Lipschitz coefficient, the existenve and uniqueness of adapted solution is established too.Lemma 1.2.2. Let (Yt,Zt, Kt)0?t?T be a solution of RBSDE (1.2) with data (?,f, T). Then there exists a constant C?depending only on?such thatTheorem 1.2.5. Let?>0 large enough and a=(at)t?0 a nonnegative Ft-adapted process. Assume g/a?H2(?, a) and (A1.4)-(A1.5) hold. Then RBSDE (1.2) with data (?, g,S) has a solution.Consider the following RBSDE driven by a Levy process: We first derive a priori estimate of solution of RBSDEL(1.9):Lemma 1.3.5. Let?>0 large enough and assume (A1.2.1)-(A1.2.5) hold. Let (Yt,Zt, Kt)0?t?T be a solution of RBSDEL (1.9) with data (?,f, T). Then there exists a constant C?depending only on 3 such that Then, we proved the existence and uniqueness of adapted solution for a simplified version of RBSDEL (1.9).Theorem 1.3.6. Let?>0 large enough and a=(at)t?0 a nonnegative Ft-adapted process. Assume g/a?H2(?,a) and (A1.4)-(A1.5) hold. Then RBSDEL (1.9) with data (?, g, S) has a solution.Proposition 1.3.7. With the same assumptions of Theorem 1.3.6, the RBSDEL (1.9) with data (?,g,S) has at most one solution. Furthermore, we haveTheorem 1.3.8. Assume (A1.2.1)-(A1.2.5) hold for a sufficient large?>0, Then RBSDEL (1.9) with data (?,f, S) has a unique solution.Chapter 2:We first provide some a priori estimates of solution of RBDSDE, then we derive the existence and uniqueness of Lp-solution of RBDSDE with Lipschitz assumption for p?(1,2).Lemma 2.3.1. Assume (A2.1)-(A2.3) hold.let (Y,Z,K) be a solution of the RBDSDE (2.1). If Y?Sp then Z?Mdp and there exists a constant k>0 such thatLemma 2.3.2. Assume that (H1)-(H3) hold, let (Y, Z, K)be a solution of the RBDSDE (2.1), where Y?Sp. Then there exists a constant k>0 such that Theorem 2.3.5. Assume (A2.1)-(A2.4) hold. Then RBDSDE (2.1) admits a unique solution (Y,Z,K)?Sp×Mdp×Scip.Chapter 3:In this Chapter, we study the backward stochastic Volterra integral equation driven by a Levy process (BSVIEL in short). We prove the existence and uniqueness of M-solution under Lipschitz assumption. Then a duality principle and a comparison theorem as well as a stable result are established. At the end of this Chapter, we show an application of M-solution for BSVIEL.Theorem 3.4.1. Let (A3.3.1) holds. Then for any?(·)?LFT2(0,T;R), BSVIEL (3.1) admits a unique M-solution (Y(·), Z(·,·), U(·,·))?LF2(0,T;R)×L2(0,T; LF2(0,T;Rd))×l2(0,T;LF2(0,T;R)).We have the following duality principle.Theorem 3.5.1. Assume (A3.4.1) holds. Let X(t) be the solution of the following R-valued forward stochastic Volterra integral equation: Then, we have the following duality principle:Theorem 3.5.3. Let f, f:[0,T]2×R×Rd×l2?R satisfying (A3.3.1) and (A3.4.2), let?>(·),?(·)?LFT2(0,T;R) such that and We denote by (Y(·), Z(·,·), U(·,·)) (resp. (Y(·), Z(·,·),U(·,·)))be the adapted M-solution to BSVIEL (3.38) corresponding to (f,?) (resp. (f,?)). ThenChapter 4:We study a class of backward doubly stochastic Volterra integral equation driven by a Levy process (BDSVIEL in short). We prove the existence and uniqueness of adapted solution to the simplified version of BDSVIEL (4.17). Based on this, we prove the existence and uniqueness of S-solution of BDSVIEL (4.17) in general case.Lemma 4.4.3. Suppose (A4.1) holds. Then, for any (t, s)??[S, T],?(·)?LFT2[S, T], the equation BDSVIEL (4.17) has a unique adapted S-solution (Y(·), Z(·,·))?*H2[S,T].Lemma 4.4.4. Suppose (A4.1) holds. Then, for any t?[R,S],?(·)?LFT2[R,S], the equation BDSVIEL (4.19) has a unique adapted solution (?S(·),Z(·,·),U(·,·))?LFS2[R,S]×L2(R,S;LF2[S,T])×l2(R,S;LF2(R,S)).Theorem 4.4.5. Suppose (A4.1) holds. Then, for any (t,s)??[S,T]?(·)?LFT2[S,T], the equation (4.1) has a unique S-solution (Y(·), Z(·,·))?* H2[S,T].