随机微分方程改进的分裂步单支θ方法的强收敛性

STRONG CONVERGENCE OF THE IMPROVED SPLIT-STEP ONE-LEG θ METHODS FOR STOCHASTIC DIFFERENTIAL EQUATIONS

本文提出了一个改进的分裂步单支θ方法,在漂移项系数满足单边Lipschitz条件下,证明了当数值方法的参数θ满足1/2≤θ≤1时,该数值方法对于这类随机微分方程是强收敛的,并在现有文献的基础上将方法的收敛阶从1/2阶提高到1阶;当0≤θ≤1/2时,若漂移项系数进一步满足线性增长条件,该数值方法也是强收敛的,收敛阶为1阶.文末的数值试验验证了理论结果的正确性.

In this paper a class of methods, called improved split-step one-leg theta methods(ISSOLTM), are introduced and are shown to be convergent for SDEs with one-sided Lipschitz continuous drift coefficient if the method parameter satisfies 1/2 ≤θ≤1. At the same time, we improve the strong order from one half to one on the basis of the existing literature.For 0≤θ≤1/2, under the additional linear growth condition for the drift coefficient, the methods are also strongly convergent with the the order 1. Finally, the obtained results are supported by numerical experiments.