分段连续型随机微分方程的均方稳定性分析(英文)

Mean square stability of stochastic differential equations with piecewise continuous arguments

考虑了自变量分段连续型随机微分方程dX(t)=(a1X(t)+a2X([t]))dt+(b1X(t)+b2X([t]))dW(t)的解析解和数值解的均方稳定性.得到了解析解的表达形式,证明了当2a1+b12+b22+2|a2+b1b2|<0时,解析解是均方稳定的.在此条件下,讨论了由半隐式欧拉方法得到的数值解的稳定性,得到如下结论:当0≤θ<a1-|a2|2a1时,0<h<-2a1+b12+b22+2|a2+b1b2|(|a1|+|a2|)((1-2θ)|a1|+|a2|);当a1-|a2|2a1≤θ≤1,0<h<∞.

Abstract： The mean - square stability of the analytic and numerical solutions of linear stochastic dif- ferential equations dX（t） = （a1X（t） ＋a2X[（t] ） ）dt ＋ （b1X（t） ＋ b2X（[t] ） ）dW（t） with piecewise continuous arguments is investigated. The explicit form of the analytic solutions is obtained, and it is proved that under the condition 2a1 ＋ b^21 ＋ b^22 ＋ 2｜a2 ＋ b1b2｜ 〈 0, the analytic solutions are mean -square stable.The semi -implicit Euler method is defined, and the mean -sqaure stability of the numerical solutions is discussed under the condition. The result is when 0〈h〈-2a1＋b^21＋2｜a2＋b1b2｜（｜a1｜＋｜a2｜）（（1-2θ）｜a1｜＋｜a2｜）;当a1-｜a2｜2a1≤θ≤1,0〈h〈∞.