欧氏空间和函数空间中的倒向随机方程

【作者】 陈绍宽；

【导师】 汤善健；

【作者基本信息】 复旦大学 ， 运筹学与控制论， 2010， 博士

【摘要】 本文研究倒向随机微分方程,倒向重随机微分方程以及由Brown运动和(?)oisson点过程一起驱动的倒向随机积分偏微分方程.全文分为两部分.第一部分讨论倒向随机微分方程以及倒向重随机微分方程的Lp解(1<p≤2).在第二章,利用光滑函数序列来逼近生成元,我们首先证明了当生成元f不依赖于第一个未知元y,关于第二个未知元z一致连续且线性增长时,一维倒向随机微分方程存在唯一的L2解.其次：通过对生成元的单调逼近,我们证明了当生成元f关于未知元(y,z)一致连续且线性增长,关于第一个未知元y的连续模具有一定的可积性时,一维的倒向随机微分方程存在唯一的Lp解.在第三章,利用弱收敛的方法,我们首先证明了当漂移系数f关于第一个未知元y单调连续且具有相当一般的增长性,关于第二个未知元z Lipschitz连续,和扩散系数g关于未知元(y,z) Lipschitz连续且关于第二个未知元z的Lipschitz系数小于1时,倒向重随机微分方程存在唯一的L2解.然后我们证明了含有正向和倒向It6随机积分的It6公式.最后,通过建立先验估计,我们得到当漂移系数f关于第一个未知元y单调连续且满足更一般的可积性条件,关于第二个未知元z Lipschitz连续,和扩散系数g关于未知元(y,z) Lipschitz连续且关于第二个未知元z的Lipschitz系数小于(?)时,多维倒向重随机微分方程存在唯一的Lp解.第二部分用概率的方法讨论倒向随机积分偏微分方程的经典解的存在唯一性,这类方程来源于对由Brown运动和Poisson点过程一起驱动的非Markov正倒向随机微分系统的研究.在第四章,我们首先证明了由Brown运动和Poisson点过程一起驱动的半鞅所对应的Ito-Wentzell公式.其次,借助于Galerkin逼近以及压缩映照原理,我们得到了非退化的随机发展方程解的存在唯一性.将这一抽象结果应用到具体的方程并利用逼近的技巧,我们进一步得到了退化的随机积分偏微分方程解的存在唯一性.再结合已建立的Ito-Wentzell公式,我们导出了由Brown运动和Poisson点过程一起驱动的随机微分方程解的逆映照所满足的随机积分偏微分方程.接着,我们对带有Poisson跳的倒向随机微分方程的解进行正则性分析.最后,通过复合倒向随机微分方程的解和正向随机微分方程解的逆映照,我们构造出倒向随机积分偏微分方程的经典解.从而我们得到了随机Feynman-Kac公式.更多还原

【Abstract】 The thesis is concerned with backward stochastic differential equations (BSDEs, for short), backward doubly stochastic differential equations (BDSDEs, for short), and semi-linear systems of backward stochastic integral partial differential equations driven by both Brownian motions and Poisson point processes. It consists of two parts.The first part is concerned with the Lp solutions (1< p< 2) to BSDEs and BDSDEs. In Chapter 2, via approximating the generator by smooth functions, we first show that there is unique L2 solution to a one-dimensional BSDE if the generator f is independent of the first unknown variable y, uniformly continuous with respect to the second unknown variable z, and of linear growth. Then, by approximating the generator by monotone functions, we prove that there is unique Lp solution to a one-dimensional BSDE if the generator f is uniformly continuous with respect to the unknown variables (y,z) and is of linear growth, and if the modulus of continuity of f with respect to the first unknown variable y is of certain integrability. In Chapter 3, by means of weak convergence, we first prove that there is unique L2 solution to a multi-dimensional BDSDE if the drift coefficient f is monotone, continuous, and of a rather general growth in the first unknown variable y, and is Lipschitz continuous in the second unknown variable z, and if the diffusion coefficient g is Lipschitz continuous in the unknown variables (y, z) and moreover, the Lipschitz coefficient with respect to the second unknown variable z is less than 1. Then we prove an Ito formula which involves both forward Ito integral and backward Ito integral. Finally, by establishing some a prior estimates, we show that there is unique Lp solution to a multi-dimensional BDSDE if the drift coefficient f is monotone, continuous, and of a more general integrability in the first unknown variable y, and is Lipschitz continuous in the second unknown variable z, and if the diffusion coefficient g is Lipschitz continuous in the unknown variables (y, z) and moreover, the Lipschitz coefficient with respect to the second unknown variable z is less than (?)The second part is concerned with, from a probabilistic point of view, semi-linear systems of backward stochastic integral partial differential equations, which arise from the study of non-Markovian forward-backward stochastic differential equations driven by both Brownian motions and Poisson point processes. In Chap-ter 4, we first establish an Ito-Wentzell formula for the semimartingales driven by both Brownian motions and Poisson point processes. Then we obtain the existence and uniqueness of solutions to non-degenerate stochastic evolution equations via Galerkin approximation and the contraction mapping principle. Applying the ab-stract result to concrete equations and using the skill of approximation, we further prove the existence and uniqueness of solutions to degenerate stochastic integral partial differential equations. Combining the Ito-Wentzell formula, we derive the stochastic integral partial differential equation for the inverse of a stochastic flow generated by a stochastic differential equation driven by both Brownian motion and Poisson point process. Then we analyze the regularity of the solutions to the BSDEs with Poisson jumps. Finally, by the composition of the random field generated by the solution of a backward stochastic differential equation with the inverse of the stochastic flow stated above, we construct the classical solution to the system of backward stochastic integral partial differential equations. As a result, we establish a stochastic Feynman-Kac formula.更多还原