An application of Taylor series in the approximation of solutions to stochastic differential equations with time-dependent delay
The authors propose an analytic approximation method for solving stochastic differential equations with time-dependent delay: \[ \begin{aligned} dx(t) & =f(x(t),x(t-\delta(t)),t)dt+f(x(t),x(t-\delta(t)),t)dw(t),\;t\in [ t_{0},T],\\ x(t_{0}) & =\{\xi(\theta),\;\theta\in[-\tau,0]\},\\ \delta & :\;[t_{0},T]\longrightarrow[0,\tau]. \end{aligned} \] The method is based on Taylor approximations of the coefficients \(f\) and \(g\). The authors consider the \(L^{p}\) convergence and the convergence with probability one of the approximate solutions.
stochastic differential equations with time-dependent delay
scientific article; zbMATH DE number 5918669