Properties Study For The Adapted Solutions About Some Terminal Or Boundary Value Problem Of Differential Equations Driven Multiply By Stochastic Processes

The purpose of this dissertation is to investigate the properties of the adapted solutions of several kinds of terminal value problems and boundary value problems of stochastic differential equations driven by three mutually independent processes:two continuous Wiener process Wt, Bt, and a Levy process Lt. The corresponding?-algebra Ft is neither increasing nor decreas-ing, so it does not constitute a general filtration. There are three main contributions. Firstly, by proving the Ito formula and Doleans-Dade formula of the Skorohod integral with jump in the sense of general Skorohod integration, we show a comparison theorem of the backward doubly stochastic differential equation driven by Levy process when the coefficient function f is de-pendent on Z, and the sizes of all negative jumps are uniformly small. To our knowledge, this is the first time that such theorem is suggested. Secondly, we show the existence and uniqueness of the adapted solution of backward stochastic differential equations driven by Levy process under various Bihari conditions. Lastly, different from the martingale representation theorem of Pardoux and Peng, by introducing the idea of variational approximation the existence of the adapted solution of the two-point boundary value problem of the stochastic differential equa-tion is investigated, for which the sufficient and necessary condition is indicated. Meanwhile, an adapted solution of the underlying equation is obtained by constructing a solution sequences. The dissertation is organized as the following.Chapter 1 gives some preliminaries for this dissertation, including some results and prop-erties of Levy process, Skorohod integral and Malliavin differential.In Chapter 2, we study the backward doubly stochastic differential equations driven by Levy process: which have important applications in finance theory and the probability interpretation of partial differential equations. By discussing the Malliavin differentiability of the solution of the above equation, we draw a comparison theorem of which the assumed conditions can be checked very easily. By methods in approximation of functions, Section 2.1 proves the existence(Theorems 2.3.1 and 2.3.2) of the higher-order moments of the solutions of the backward doubly stochastic differential equations when the terminal value?and the coefficient functions satisfy certain con-ditions. According to the properties of the Malliavin differential and Picard iteration, Sections 2.2 and 2.3 prove the existence of the first order and second order Malliavin derivatives of the backward doubly stochastic differential equation with respect to the Wiener process{Bt}, re-spectively. One-order and two-order Malliavin derivatives of the solutions are proved to be sat-isfied with a linear backward doubly stochastic differential equation(Theorems 2.4.1 and 2.5.1). Meanwhile, the existence(Theorems 2.4.2 and 2.5.2) of higher-order moments of the first or-der and second order Malliavin derivatives of the solutions of the backward doubly stochastic differential equations are proved respectively. In Section 2.4, the Ito formula(Theorem 2.6.1) and Doleans-Dade formula(Theorem 2.6.2) of the Skorohod integral with jump are proved, the Levy process from which the Teugels martingales are derived is discussed thoroughly. After that, we draw a new comparison theorem(Theorem 2.6.4) for the adapted solution in the sense of that the sizes of all negative jumps are uniformly small when the coefficient function f is dependent on Z.The third chapter discusses the existence and uniqueness of the adapted solution of the backward stochastic differential equation driven by Levy process under various Bihari condi-tions. In Section 3.1, we consider the backward doubly stochastic differential equations driven by Levy process(Eq.(1)). When the coefficient g satisfies the Lipschitz condition, and f satis-fies the generalized Bihari condition: |f(t,y1,u1,z1)-f(t,y2,u2,z2)|2?< c(t)k(|y1-y2|2)+K(|u1-u2|2+?z1-z2?2), we can prove the existence and uniqueness(Theorem 3.2.1) of the Ft-adapted solutions of Eq.(1) by the generalized Ito's formula, Picard iteration, and interval extension process. By the generalized Bihari inequality and truncation function, the existence and uniqueness(Theorem 3.3.2) of the solution of the backward tochastic differential equation driven by Levy process under local Bihari conditions are proved in Section 3.2. By the generalized Bihari inequality again and some smooth functions, Section 3.3 proves the same results(Theorem 3.4.1 and 3.4.2) but under monotony Bihari conditions.By introducing the idea of variational approximation, Chapter 4 investigates the sufficient and necessary conditions(Theorem 4.3.1) for the existence of the adapted solutions of the two-point boundary value problems of the stochastic differential equation of the following form dXt=f(t, Xt)dt+?(t, Xt)dWt. AX0+BXT=?*. In the simple case of f(t, Xt)=ft, a solution of this equation(Theorem 4.4.1) can be obtained by introducing a control term ft, extending the solution from Xt to (Xt,ft), and construct-ing a solution sequence. Meanwhile, we prove the continuous dependency(Theorem 4.5.1) of the constructed solution on the boundary value. Lastly, the sufficient and necessary con-ditions for the solution we obtained are compared with those of the Strum-Liouvelle prob-lems(Example 4.6.1), and the "martingale approximation" problem(Example 4.6.2) proposed by Peng for backward stochastic differential equations, which show the universality of our variational adapted solution.Chapter 5 gives the conclusion and future work.