A Note on Single-Step Difference Scheme for the Solution of Stochastic Differential Equation

This is a discuss on the application of operator approach to stochastic partial differential equations with dependent coefficients. Single step difference schemes generated by exact difference scheme for an abstract Cauchy problem for the solution of stochastic differential equation in a Hilbert space with the time-dependent positive operator are presented. The main theorems of the convergence of these difference schemes for the approximate solutions of the time-dependent abstract Cauchy problem for the parabolic equations are established. In applications, the convergence estimates for the solution of difference schemes for stochastic parabolic differential equations are obtained. Numerical results for the ${1}/{2}$ order of accuracy difference schemes of the approximate solution of mixed problems for stochastic parabolic equations with Dirichlet and Neumann conditions are provided. Numerical results are given.