Backward Stochastic Differential Equations With Non-Lipschitz Coefficients And The Related Research

The foundation paper of general nonlinear backward stochastic differential equa-tion(BSDE in short) was given by Pardoux and Peng[97]. BSDEs are equations defined on [0, T] and having the following type: (1.1) g(t,y,z) is said to be the generator of BSDE(1.1), (T,ξ) is said to be the terminal conditions of BSDE(1.1). Now it is well known that equation (1.1) has a pair of unique adapted and square integrable solution providing that the generator g(t, y, z) is Lips-chitz in both variables y and z, and thatξand (g(t,0,0))t∈[0,T] are squre integrable. From then on, a larg amount of works have been devoted to the study of BSDE theory. The first important one is the fundamental research including to establish the existence and uniquness of solutions to BSDE(1.1) under more complex forms(e.g. BSDEs with jumps; FBSDEs; BSDEs with reflected; BSDEs dieven by martingale; et al.) and/or un-der weaker coefficients assumptions for extending Pardoux-Peng's initial result. One can refer to Pardoux-Peng[101], El Karoui[41], Lepeltier-San Martin[79; 80], Kobylanski[77], Briand-Hu [10; 11], Chen[27], Jia[62; 63; 65], Briand-Delyon-Pardoux-Hu-Stoica[9], Mao[92], Hu-Peng[59], Hu-Yong[60], Peng-Wu[115], Ma-Protter-Yong[88], Ma-Yong[89], Pardoux-Tang[99], Peng-Shi[116], Wu[121; 122; 124], EI Karoui-Kapoudjian-Pardoux-Peng-Quenez[45], Kobylanski-Lepeltier-Quenez-Torres[78], Matoussi[93], Hamadene-Lepeltier-Matoussi[49], Hamadene[51], Hamadene-Lepeltier-Wu[49], Lepeltier-Matoussi-Xu[81] and the refrences therein. This part also includes the study of the various properties of BSDE(1.1)(e.g. the Comparision Theorem; the Converse Comparision Theorem; the Representation Theorem; the Jessen's inequality; et al.). e.g. see Peng[104; 105], El Karoui-Peng-Quenez[42], Briand-Coquet-Hu-Memin-Peng[8], Coquet-Hu-Memin-Peng[31], Cao-Yan[14], Liu-Ren[85], Wu[123], Chen-Kulperger-Jiang[20; 21], Jiang[66; 67; 68; 70; 71], Jia[65] and the refrences therein. At the same time, with the deepening of the study, it was found that BSDE theory could be applied and became very powerful tool in many fields. To be specific, BSDE theory can be used in stochastic opotimal control and stochastic games (e.g. see Peng[104; 106; 108; 109], Hamadene-Lepeltier-Peng[52], Hamadene-Lepeltier-Wu[49], Hamadene[50], Lim-Zhou[82], Kohlmann-Tang[74; 75], Kohlmann-Zhou[76], Liu-Peng[87] et al.) and in the theory of hedging and nonlinear pricing theory for incomplete markets (e.g. see El Karoui-Peng-Quenez[42], Karoui-Quenez[44],Chen-Epstein[18], Duffi-Epstein[38], Cvitanic-Karatzas[34], Yong[125] et al.) and in nonlin-ear expectation theory (e.g. see Peng[105; 107; 111; 112], Briand-Coquet-Hu-Memin-Peng[8], Coquet-Hu-Memin-Peng[29], Chen-Peng[26], Chen-Epstein[18], Chen-Chen-Davison[17], Chen-Kulperger-Jiang[20; 21], Chen-Wang[24], Jiang[66], Rosazza[118] et al.). The BSDE theory also provides probabilistic formulae for solutions to partial differential equations(e.g. see Peng[102], Pardoux[95], El Karoui-Kapoudjian-Pardoux-Peng-Quenez[45], Pardoux-Peng[101], Barles-Buckdahn-Pardoux[4], Pardoux-Tang[99], Buckdahn-Hu[13], Kobylanski[77] et al.).Since the frst existence and uniqueness result, many papers have been devoted to existence and/or uniqueness results under weaker assumptions. Among these papers, we can distinguish two different classes:scalar BSDEs and multidimensional BSDEs. In the first case, one can take advantage of the comparison theorem. For multidimensional BSDEs, there is no comparison theorem and to overcome this difficulty a monotonicity assumption on the generator f in the variable y is used. Instead of using the mono-tonicity assumption, Mao[92] also proposed a kind of non-Lipschitz assumption to deal with the multidimensional BSDEs and with the help of Bihari's inequality, he proved the existence and uniquness result.However, all the assumptions mentioned above (both scalar and multidimensional) are independent of time variable t. Our dissertation is dedicated to the fundamental study of BSDE, we propose a kind of non-Lipshcitz assumption which is related to time variable t to the generator of BSDE (1) and get the existence and uniqueness result for the adapted solution of equation (1). This existence and uniqueness result is also the theoretically basis of the whole thesis, by which we disicuss some related problems that play important roles in the applications.The dissertation includes the following chapters: Chapter 1 BSDEs with non-Lipschitz Coefficients;Chapter 2 The Monotonic Limit Theorem and Nonlinear Decomposition Theorem of g-supermartingale;Chapter 3 Representation Theorem for Generators of BSDE;Chapter 4 Adapted Solution of a Backward Semilinear Stochastic Evolution Equation with Non-Lipschitz Coefficients.(I) In Chapter 1, we give the following non-Lipschitz assumption:g satisfies (H1.3) where c>1 and p(t, u) satisfies:for fixed t∈[0, T],ρ(t,·) is a continuous concave nondecreasing function such thatρ(t,0)=0.for fixed u,the following ODE has a unique solution u(t)=0, t∈[0,T].there exits a(t)≥0, b(t)≥0, such thatρ(t, u)≤a(t)+b(t),, andThe main relult is Theorem 1.3.1Theorem 1.3.1 Letξ∈L2(Ω,FT, P) and assume that (H1.2) and (H1.3) hold, then the BSDE (1.1) has a unique solution (yt, zt).We also get the comparison theorem of equation (1). Theorem 1.4.1 Let (y. z), (y, z) be the unique solution of the following BSDEs whereξ,ξ∈L2(Ω,,FT,P), g(t,y,z),g(t,y,z) satisfy (H1.2) and (H1.3). If (i)ξ≤ξa.s., (ii) g(t, y, z)≤g(t, y, z) a.s., a.e.. Then we have Moreover, if we assume P(ξ<ξ)>0, then P(yt 0. For a family of BSDEs parameterized byε>0, (1.10) When gεsatisfies our non-Lipschitz assumption (H1.3), we get a stability theorem.Theorem 1.5.1. Under assumption (H1.6),(H1.7) and (H1.8), we have(II) In Chapter 2, we study the monotonic limit theorem and nonlinear decompo-sition theorem of g-supermartingale under our non-Lipschitz assumption. The following are the main result Chapter 2 For BSDEs we haveTheorem 2.3.2.(Monotonic Limit Theorem) We assume that g(t,y,z) satis-fies (H1.2) and (H1.3) and (Ai) satisfies (H2.1). Let (yi,zi) be the solution of BSDE (2.7), with E[sup 0≤t≤T |yti[2]<∞. If(yti) increasingly converges to ((t) with E[sup 0＜t＜T|yt|2]<∞, then (yt) is a g-supersolution. i.e., there exist a (zt)∈M2(0,T;Rd) and a RCLL square integrable increasing process (At) such that the pair (yt,zt) is the solution of the BSDE (2.8) where (zt)0≤t≤T is the weak (resp. strong) limit of{(zti)} in M2(0, T;Rd) (resp. in Mp(0, T; Rd), for p<2) and, for each t,At is the weak limit of{Ati} in L2(Ω, Ft, P).Theorem 2.4.3.(Decomposition Theorem) Assume that (H1.2), (H1.3) and (H2.3) hold. Let (Yt) be a right continuous g-supermartingale on [0, T] in strong sense with Then (Yt) is a g-supersolution on [0, T]: there exists a unique RCLL increasing process (At) with A0=0 and E(AT)2<∞such that (Yt) coincides with the unique solution (yt) of the BSDE (2.15)(III) In Chapter 3, under non-Lipschitz condition, we get the representation theorem for the generator of BSDE. With the help of the representation theorem, we prove the uniqueness for g-expectation and the converse comparison theorem for generators of BSDEs. We denote the unique solution of BSDE (1) by (yg,T,ξ,zg,T,ξ). The main result of Chapter 3 is the following theorem.Theorem 3.3.5.(Represention Theorem) Let (H1.1) and (H3.1) hold for g, 1≤p<2. Then for each pair (y, z)∈R×Rd, the following equalityholds for almost every t∈[0, T].Theorem 3.4.1 Let (H1.2) and (H3.1) hold for two genreators g1 and g2. Assum that dP×dt-a.s., g(t,0,0)≡0. Then the follwing statements are equivalent: (i) dP×dt-a.s., (?)(y, z)∈R×Rd, g1(t, y, z)=g2(t, y, z). (ii)y0(g1,t,ξ)=y0(g2,t,ξ),(?)t∈[0,T],ξ∈L2(Ω,Ft,P).Theorem 3.4.3 Let (H1.2) and (H3.1) hold for two genreators g1 and g2.Then the follwing statementsare equivalent: (i) dP×dt-a.s., (?)(y,z)∈R×Rd, g1(t,y,z)≥g2(t,y,z). (ii)(?)t∈[0,T],ξ∈L2(Ω,Ft,P), P-a.s., ys(g1,t,ξ)≥ys(g2,t,ξ),(?)s∈[0,t]. (iii)(?)t∈[0,T],(y,z)∈R×Rd,ε∈[0,T-t],(IV) In Chapter 4, we study the backward semilinear stochastic evolution equation in infinite dimensional space. (4.1) The main result of this Chapter is the following existence and uniqueness adapted solution for equation (4.1) with non-Lipschitz assumption.Theorem 4.2.4. Assume that X∈L2(Ω,FT,P;H) and (H4.1)and (H4.2) hold true. Then there exists a unique pair (x(·),y(·))∈M2(0,T;H)×M2(0,T;L2(K,H)) satisfies equation (4.2).