AIMS Mathematics, 2017, 2(3): 377-384. doi: 10.3934/Math.2017.3.377.

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

1 Department of Mathematics, Tokyo Gakugei University, Koganei, Tokyo, 184-8501, Japan
2 Department of Mathematics, Yokohama National University, Hodogaya, Yokohama, 240-8501, Japan

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.
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Keywords Difference equation; weakly dependent random variables; Euler-Maruyama scheme; strong invariance principle; linear stochastic di erential equation

Citation: Hiroshi Takahashi, Ken-ichi Yoshihara. Approximation of solutions of multi-dimensional linear stochastic differential equations defined by weakly dependent random variables. AIMS Mathematics, 2017, 2(3): 377-384. doi: 10.3934/Math.2017.3.377


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