分数随机微分方程的定性理论研究及其应用

All kinds of fractional operators and the corresponding fractional differential equations appeared more and more frequently in almost all of the research fields and the engineering applications, as the development of the fractional operators theory. Fractional stochastic differential equations, especially fractional stochastic delay differential systems driven by fractional Brownian motion, occupied an important position in many fields such as system identification, option pricing, finance and insurance, system control. And the corresponding theoretical research results can can be applied in the related fields directly or indirectly. This project aim at the fractional stochastic differential system, the main contents include: (1) We mainly discuss the solvability of the fractional stochastic differential equations. By means of a powerful tool in the nonlinear analysis and the stochastic skills, some properties of mild solution for the equations will be considered. (2) We will study the numerical solution for the fractional stochastic differential equations. This study on qualitative theory will be very difficult because of the invalid Itô formula. It is novel that this project consider the stablility of system from the perspective of numerical solution. (3) Finally, Poyang lake ecological wetland migratory birds population system will be studied by Lotka-Volterra equations. This project can provide some scitific basis and positive reference by considering the stability of the population system.

随着分数阶算子理论的发展，各类分数阶算子及相应的分数微分方程越愈加繁地出现于几乎所有的研究领域和工程应用之中。分数随机微分方程特别是由分数布朗运动驱动的分数随机时滞微分系统在系统识别、期权定价、金融保险、系统控制等领域中占据重要位置，其理论研究成果能直接或间接地服务于相关领域。本项目以分数随机微分系统为研究对象，主要内容包括：(1)研究分数随机微分方程的可解性，通过非线性分析理论及随机分析技巧，考虑其mild解的存在性、唯一性等性质。(2)研究分数随机微分方程的数值解情况。对于分数随机微分方程而言，由于Itô公式已不再适用，导致对定性理论的研究难度加大。本课题从数值解的角度来研究系统的稳定性情况，从研究角度来说独树一帜。(3)利用随机Lotka-Volterra方程对鄱阳湖湿地生态的候鸟种群进行模型的刻画，通过考虑种群系统的稳定性，从而对鄱阳湖生态湿地的保护与发展提供一定的科学依据。

本课题主要研究了分数阶随机微分方程的可解性与可控性等定性理论问题。通过迭合度理论，研究了一类分数阶微分耦合系统解的存在性；通过Krasnoselskii不动点定理及随机分析技巧，研究了一类分数随机微分方程的可解性与可控性。同时，项目还对分数随机微分系统的数值解及稳定性作了相关讨论。