马尔可夫调制的随机变延迟微分方程的数值解

讨论了马尔可夫调制的随机变延迟微分方程dx(t)=f(x(t),x(t-δ(t)),r(t))dt+g(x(t),x(t-δ(t)),r(t))dW(t)欧拉方法的收敛性.对方程应用欧拉方法,特别地对变延迟部分运用插值技巧进行数值离散后,将离散的欧拉格式延拓为连续的欧拉格式,从而得到欧拉格式在局部Lipschitz条件下强收敛到解析解.进一步,将局部Lipschitz条件换成全局Lipschitz条件,结论也成立,即欧拉方法在全局Lipschitz条件下也是强收敛的.

The numerical solutions of stochastic differential equations with variable delay and Markovian Switching

The convergence of the Euler method for stochastic differential equation dx(t)=f(x(t),x(t-δ(t)),r(t))dt+g(x(t),x(t-δ(t)),r(t))dW(t)with variable delay and Markovian Switching is investigated.Applying the Euler method to the equation and particularly using the interpolation technique on vailable delay.the Euler approximation is obtained.The discrete Euler approximation is extended to the continuous Euler approximation.It iS proved that the Euler numerical solutions will converge strongly to the true solution under the local Lipschitz condition.Furthermore,replacing the local Lipschitz condition by the global Lipschitz condition,the same conclusion is obtained,that iS to say,the Euler method is strong convergent under the global Lipschitz condition.