带跳的非线性随机微分方程数值解及其在Lotka-Volterra模型中的应用研究

At present, the most numerical solutions for nonlinear stochastic differential equations with jumps need the drift and diffusion coefficients satisfy the global Lipschitz condition and the linear growth condition, but there are many nonlinear models do not satisfy the above conditions in the finance, biology and other fields. Therefore, the delvelopment of the numerical methods for the equations with non-global Lipschitz contion is very meaningful. . In this project, we foucs on the numerical methods for nonlinear stochastic differential equations with jumps under the non-global Lipchitz condition and the research as follows：Firstly，several high efficient numerical methods are constructed for the nonlinear equations with super-linear drift and diffusion coefficients, i.e. the backward Euler method, split-step theta method and compensated split-step theta method; and then the convergence and its order for the algorithms are studied. Secondly, the mean square stability and almost surely exponential stability of the above numerical methods for the nonlinear equations are further discusssed. Thirdly, the above three numerical methods are applied to the numerical simulation of the stochastic Lotka-Volterra model with jumps, and the convergence of numerical solutions is studied. The research of this project can propose the effective and feasible numerical algorithms for the models of biomathematics and epidemic control.

目前关于带跳的非线性随机微分方程的数值解，大都是要求方程的漂移系数和扩散系数满足全局Lipschitz条件和线性增长条件，而在金融、生物等领域存在着许多非线性的模型不能满足上述条件，这大大限制了数值解法的应用。因此，发展非全局Lipschitz条件下该类方程的数值解是十分有意义的。.本课题基于非全局Lipschitz 条件下带跳的非线性随机微分方程数值解法进行如下研究：1、构造漂移系数和扩散系数均为超线性增长条件下的几种高效数值算法，即向后欧拉法、分步theta法和补偿分步theta法，并研究其强收敛性和收敛阶问题；2、进一步研究上述三种数值方法的均方稳定性和几乎必然稳定性；3、将以上三种数值方法用于带跳的随机Lotka-Volterra模型的数值模拟，并研究数值解的收敛性。本课题的研究成果为生物数学模型及流行病控制提出有效可行的数值算法。

目前关于带跳的非线性随机微分方程的数值解，大都是要求方程的漂移系数和扩散系数满足全局Lipschitz条件和线性增长条件，而在金融、生物等领域存在着许多非线性的模型不能满足上述条件，这大大限制了数值解法的应用。因此，本项目提出了发展非全局Lipschitz条件下该类方程的数值解的研究。本项目主要研究了在非全局Lipschitz条件下该类方程几种数值算法的收敛性和稳定性问题，并将这些数值算法应用到生物和金融系统中。本项目构造出向后欧拉法，分步theta法，隐式平衡法，理论上证明了这些算法的收敛性和稳定性，并在生物系统中的随机年龄结构模型和金融学中跳扩散模型中得到了很好的应用。本项目的研究成果可以为生物学和金融学模型的数值模拟提供了高性能的算法和理论依据。