非线性随机分数阶微分方程Euler方法的弱收敛性

THE WEAK CONVERGENCE OF EULER METHOD FOR NONLINEAR STOCHASTIC FRACTIONAL DIFFERENTIAL EQUATIONS

论文首先证明了非线性随机分数阶微分方程解的存在唯一性,然后构造了数值求解该方程的Euler方法,并证明了当方程满足一定约束条件时,该方法是弱收敛的.特别地,当分数阶α=0时,该方程退化为非线性随机微分方程,所获结论与现有文献中的相关结论是一致的;当α≠0,且初值条件为齐次时,所获结论可视为现有文献中线性随机分数阶微分方程情形的推广和改进.随后,文末的数值试验验证了所获理论结果的正确性.

This paper is concerned with the existence and uniqueness of solutions for nonlinear stochastic fractional differential equations and the weak convergence of Euler method constructed for solving the equations when they satisfy certain constraints.Especially,when fractional orderα=0,the equations are degenerated to nonlinear stochastic differential equations,and the conclusions obtained from this paper are consisted with the relevant results;whenα≠0 and the initial condition is homogeneous,the conclusions can be regarded as the generalization and improvement of linear stochastic fractional differential equations in the existing literature.Finally,numerical examples illustrate the effectiveness of the theoretical results.