Approximation of solutions of multi-dimensional linear stochastic differential equations defined by weakly dependent random variables

Abstract

It is well-known that under suitable conditions there exists a unique solution of a ddimensional linear stochastic differential equation. The explicit expression of the solution, however, is not given in general. Hence, numerical methods to obtain approximate solutions are useful for such stochastic di erential equations. In this paper, we consider stochastic difference equations corresponding to linear stochastic differential equations. The difference equations are constructed by weakly dependent random variables, and this formulation is raised by the view points of time series. We show a convergence theorem on the stochastic difference equations.

Keywords

• Difference equation| weakly dependent random variables| Euler-Maruyama scheme| strong invariance principle| linear stochastic di erential equation