线性随机微分方程的全隐式Euler方法

Full Implicit Euler Methods for Linear Stochastic Differential Equation

由于随机微分方程的全隐式Euler方法不是均方收敛的,一般认为它没有意义。然而,从运用计算机实现的角度来说几乎处处意义下的收敛和稳定比均方意义的收敛和稳定更具优势。针对线性随机微分方程,提出了一类全隐式Euler方法,证明了该方法生成的数值解几乎处处收敛,给出了该方法几乎处处稳定的充要条件。

The full implicit Euler methods for the stochastic differential equations were criticized for failing to converge to the true solutions in mean square. However, the almost sure convergence and stability were superior to the convergence and stability in mean square from the viewpoint of computer implementation. For the linear stochastic differential equations a class of the full implicit Euler methods was proposed, the numerical solution generated by it was almost surely convergent, and the necessary and sufficient condition of almost sure stability for the method were given.