Among their topics are geometric methods for stochastic dynamical systems, an averaging principle for multi-valued

stochastic differential equations driven by G-Brownian motion, Holder estimates for solutions of stochastic nonlocal diffusion equations, the Cauchy problem for a generalized Ostrovsky equation with positive dispersion, the smooth approximation of Levy processes in Skorokhod space, and error estimation on projective integration of expensive multiscale stochastic simulation.

With help from MATLAB (Mathworks, Inc., Natick, MA, USA) and Milsteins higher order method [34], which is a powerful tool for solving

stochastic differential equations, we consider the following discretized equation of model (7) at t = (k + 1)[DELTA]t, k = 0,1,...:

There are many works [1-3] about the existence and uniqueness of strong or weak solutions for the following

stochastic differential equation (denoted briefly by SDE):

For simplicity of notation, we suppose that each component of trajectory in d-dimensional space satisfies the following

stochastic differential equation (SDE):

Numerical analysis for

stochastic differential equation (SDE) has seen a considerable development in recent years.

Now, we consider the d-dimensional

stochastic differential equation:

The first efficient stability conditions for

stochastic differential equations with unbounded delays, obtained by the W-method, were presented in the paper [7].

White noise analysis created by Hida [1] is essentially a branch of infinite-dimensional calculus on generalized functionals of Brownian motion, which connected with the applications to the study of random processes and

stochastic differential equations.

The investigation of random periodic orbits at large and in specific

stochastic differential equations (SDEs) is a difficult dynamical problem [1].

When the fast processes of a continuous system are modeled by white noise--as is common for physical applications--the resulting stochastic model converges to a Stratonovich

stochastic differential equation (Wong and Zakai 1965; Papanicolaou and Kohler 1974; Gardiner 1985; Penland 2003a,b).

Stochastic differential equations have been widely applied in science, engineering, biology, mathematical finance and in almost all sciences.