线性随机延迟微分方程半隐式Euler方法的局部收敛性证明

The proof of the local convergence of the semi-implicit Euler method for a linear stochastic differential delay equation

给出了线性随机延迟微分方程解析解的几个重要不等式的详细证明,进而讨论了半隐式Euler方法的局部收敛性,应用Ito积分的性质、Doob不等式、Hlder不等式证明了在均方意义下半隐式Euler方法的局部收敛阶为1.

The detail proof of some important inequalities is given for exact solution of a linear stochastic differential delay equation. The local Convergence of the semi - implicit Euler method is discussed accordingly. By applying the properties of the ho stochastic integral, the Doob＇s inequality and the Hoelder inequality, it is shown that the local convergent order of the semi - implicit Euler method is 1 in the mean - square sense.