分数阶随机微分方程的修正隐式数值格式

Corrected implicit schemes for fractional stochastic differential equation

对H>1/2且随机积分为前向积分的分数阶布朗运动驱动的随机微分方程,为改善显式Euler格式和Milstein格式的稳定性,基于修正隐式技术构造了修正隐式Euler格式和Milstein格式,证明了修正隐式格式较显式格式有更大的稳定步长集,且在一定条件下修正隐式Euler格式是A稳定的.数值模拟显示,步长在稳定步长集内时数值格式稳定,步长在稳定步长集边界附近时误差几乎不改变,而步长在稳定步长集外时数值格式极度不稳定,从而验证了修正隐式格式在保持数值稳定性上的优越性和稳定步长集的合理性.

For H 〉 1/2 and forward stochastic integral for stochastic differential equation driven by frac- tional Brownian motion,in order to improve the stability of explicit Euler scheme and Milstein scheme, we construct corrected implicit Euler scheme and Milstein scheme based on the corrected implicit tech- nology. Then we prove these corrected implicit schemes have greater stable stepsets than explicit scheme,and in certain conditions the corrected implicit Euler scheme is A-stability. Finally, It is showen that the numerical schemes are stable when the step size is within the stable step set, the numerical er- rors are steady when the step size is near the set＇s bound, but the numerical schemes are extremely un- stable when the step size is out of the set. Thus the corrected implicit scheme have advantages of stability and the definition of stable step set is reasonable.