带跳的随机泛函微分方程及带测度值的边值问题

Stochastic differential equations and Dirichlet form theories have been widely used in many fields, such as biology, engineering, finance and so on. Their theoretical development has been very perfect. However, there are still many unsolved problems about neutral stochastic functional differential equations with jumps and boundary problems with measure data. This project will be devoted to the study of the following issues. 1. The existence, uniqueness, moment estimation, stability and numerical approximation of the solution of neural stochastic functional differential equation with finite delay and jump were considered, then the large deviation of the occupation measure was studied. 2. The Euler-Maruyama approximation for neutral stochastic functional differential equations with infinite delay and jump was researched, then the large deviation of the occupation measure considered. We tried to get the moment estimation, stability and error of the numerical solution. 3. The large deviations for the occupation time distribution of a class of asymmetric Markov processes with finite life-time were presented. Then they were applied to get the L^p independence of a class of generalized Feynman-Kac semigroups. 4. The boundary value problem, including Dirichlet boundary value, Neumann boundary value and mixed boundary value, of the second order elliptic differential operator with measure value was studied. When gauge function is infinite, the probability expression of the solution was given, and the uniqueness of solution was proved. When the measure was not zero, the probability expression of the solution to the above three boundary value problems were obtained under the framework of semi-Dirichlet form setting, and the existence and uniqueness of the solution in more general setting were gotten. The above problems have important theoretical and practical significance.

随机微分方程及狄氏型理论近年来被广泛应用于生物、工程、金融等领域，其理论框架已基本完善，然而带跳的中立随机泛函微分方程及带测度值的边值问题尚有许多问题需要研究。本项目将研究以下问题：1.有有限延迟的带跳的中立随机泛函微分方程解的存在唯一性、矩估计、渐近稳定性、占有测度的大偏差及EM数值逼近。2.有无限延迟的带跳的中立随机泛函微分方程EM数值解的矩估计、渐近稳定性及其与真实解的误差及解的占有测度的大偏差。3.生命时有限的非对称马氏过程占有时分布的大偏差及广义Feynman-Kac半群的Lp独立性。4.带测度值的二阶椭圆微分算子的边值问题。对测度为零的情形，当Gauge无限时，将给出（Dirichlet、Neumann、混合）边值问题解的概率表示，并证明解的唯一性；对测度不为零的情形，给出半狄氏型框架下三种边值问题解的概率表示及一般意义下解的存在唯一性。以上问题有着重要的理论意义和实际意义。

本项目研究了随机微分方程(包括随机泛函微分方程、脉冲随机微分方程、随机偏积分-微分方程、分布依赖的随机微分方程)的正则性、数值逼近、稳定性、遍历性、大偏差、边值问题、凸区域上反射问题等。完成了项目的计划要求，执行情况良好。主要成果如下：1）、对一类带跳的中立随机泛函微分方程，在更一般的条件下得到了解的存在唯一性和矩估计。对有固定延迟的带Levy跳的中立随机微分方程，得到了数值解稳定于真实解。对一类有有限延迟的随机泛函微分方程，得到了带随机步长的EM数值解稳定于真实解，给出了收敛速度。2）、给出了一类脉冲随机泛函微分方程的解依分布稳定的充分条件；对一类带扰动的随机脉冲泛函微分方程，给出了扰动方程的解和原方程的解逼近的充分条件。3）、得到了一类含有随机偏积分微分算子A的倒向随机偏微分方程弱解的存在唯一性，且A满足某种条件时，方程的解有类似于Feymann-Kac公式的概率表达式，证明了解与带跳的倒向随机微分方程有某种一一对应关系。4）、得到了两类系数非正则的分布依赖的随机微分方程弱解的存在性。5）、建立了在凸区域中反射的拟线性随机偏微分方程组解的存在唯一性。在参数满足非退化和退化条件下，得到了布朗运动驱动的一类倒向随机偏微分方程在凸区域上反射的解的存在唯一性。6）、在系数满足一定Dini连续性的条件下，证明了变阶的类稳定过程的弱唯一性。7）、给出了取值于半正定对称矩阵空间上保守仿射过程存在唯一不变分布的充分条件；对典则状态空间上一类仿射过程，当扩散项满足Hormander型条件及边界不可达时，得到了转移密度的存在性和可微性。8）、用弱收敛方法得到了一类带有快慢尺度的随机偏微分方程中慢过程的Freidlin-Wentzell 大偏差。9）、给出了一类整数序列乘积的上下界、一类独立随机变量和的概率估计、Rademacher和的绝对值小于1的概率的表达式。10）、培养了4名硕士研究生。

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