随机延迟微分方程的全隐式Euler方法

研究随机延迟微分方程数值解具有重要的意义,目前已有显式和半隐式两种数值方法,还没有全隐式的数值方法.本文构造了一种全隐式Euler方法,在该方法中用一些截断的随机变量代替维纳过程增量△W<,n>,接着证明了全隐式方法是1/2阶收敛的并通过数值实验验证了该方法的收敛性.最后,用数值实验表明在某些情况下全隐式方法的稳定性比半隐式方法好一些.

FULL IMPLICIT EULER METHOD FOR STOCHASTIC DELAY DIFFERENTIAL EQUATIONS

It is very important to study the numerical solution of the stochastic delay differential equation, now there exist the explicit and the semi-implicit numerical methods and does't exist the full implicit numerical methods. In this paper, a class of the full implicit Euler method is given. In the full implicit schemes increments of Wiener processes are substituted by some truncated random variables, then, we prove that the method converges with the order 1/2 and the property of convergence of the methods is verified by the numerical experiments. Finally, the numerical experiments show the property of stability of the full implicit methods is better than that of the semi-implicit methods under some situation.