关于几类非线性随机延迟微分系统数值解的研究

It is well known that stochastic system, has come to play an important role in science and engineering fields. Most of real world models are nonlinear, such as, population models, neural networks. Explicit solutons can rarely be obtained for highly nonlinear stochastic delay differential equations. So the numerical approximation for nonlinear stochastic delay differential equations has received more and more attention. Recently, several authors have devoted to strong convergence of implicit Euler-Maruyama schemes for nonlinear SDDEs under some.specific conditions. Thus, this project proposes “Numerical Approximation of Several Nonlinear Stochastic Delay Differential Equations” to solve several problems of this research area that has been widely concerned: To study the highly nonlinear stochastic differential equation with time-dependent delay, After providing the existence-and-uniqueness of the exact solution under the one-side polynomial growth condition and local Lipschitz condition, we prove that the backward Euler-Maruyama numerical method can preserves boundedness of moments, and the implicit numerical approximation converges strongly to the true solution. Moreover, to establish the strong convergence and stability of the Euler-Maruyama numerical solution for nonlinear stochastic differential equations with time-dependent delay and Markovian switching, and consider several highly nonlinear examples to illustrate our theory, which imply the results in the project are very general and can cover a wider class of nonlinear SDDEs. Finally, in order to study the numerical solution of stochastic function differential equation, that is more widely applied and complicated, we compare with population models and try to establish more general nonlinear conditions to replace polynomial growth condition and local Lipschitz condition, on the basis of the study of these problems.

众所周知，随机系统在许多科学和工程领域中扮演着极为重要的角色。大部分现实模型都是非线性的，只有极少的非线性随机微分方程可以求出显式解，故而非线性随机微分方程数值解这一方向获得了许多的关注。近几年来，学者们在此领域里，提出了利用各种隐式Euler数值方法所得到数值解的强收敛性结论。因此，本项目提出“关于几类非线性随机延迟微分系统数值解的研究”来研究此领域上的一些热点问题：首先，研究具有随时间变化延迟的非线性随机微分方程，提出原方程系数满足局部Lipschitz条件和多项式增长条件，建立后退Euler数值法且证明数值解可强收敛到方程理论解。另外，尝试将数值解的强收敛性和稳定性推广到有Markovian调制的变延迟非线性随机微分方程与中立型随机微分方程中。最后，在前期的研究基础之上，尝试提出更为一般性的条件去代替非线性随机延迟方程中需要的多项式条件，从而研究应用更为广泛的随机泛函微分方程数值解。

在项目执行过程中，研究内容以随机微分方程及其数值解，随机最优控制系统，非经典扩散方程，等为主要研究方向。项目成员们阅读了大量各类相关文献，在研究初期，参加国内的一些学术交流活动，与各领域的专家学者们交流，持续积累研究经验，后期与其他学者合作，在如下几个问题上有了收获，正式发表了文章。其一，针对随机广义耦合里卡蒂方程，我们通过奇异值分解，建立了其解的存在性，将此结论应用到带马尔科夫跳的线性随机奇异系统的最优控制系统中，并得到了该最优控制所期望的显式表达。其二，在可由带混合初值条件的平均场正倒向随机微分方程描述的近似最优控制问题上，利用Ekeland变分法原理与递归方法建立了Pontryagin形式的最优控制充分必要性条件，且在控制域的凸性限制下得到了结果。其三，针对非经典扩散方程，给出了半群的指数吸引子对参数的连续性的一个新的抽象判断方法，且在该抽象结果的主要假设条件中没有强加任何的紧嵌入。最后，在矩阵非阿基米德随机赋范空间中，利用直接法和不动点方法研究了一类泛函方程的Hyers-Ulam稳定性。