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AIMS Mathematics

2017, Volume 2, Issue 3: 377-384. doi: 10.3934/Math.2017.3.377
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Research article

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
  • Received: 30 October 2016 Accepted: 23 May 2017 Published: 22 June 2017

  • 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.

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

    Related Papers:

  • 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.


    加载中
    [1] P. E. Kloden and E. Platen, Numerical solution of stochastic differential equations, Springer-Verlag, Berlin Heidelberg, 1992.
    [2] W. Liu and Z. Lin, Strong approximation for a class of stationary processes, Stoch. Proc. Appl., 119 (2009), 249-280.
    [3] X. Mao, Stochastic differential equations and applications, second edition. Horwood Publishing Limited, Chichester, 2008.
    [4] H.Takahashi, S. Kanagawa and K. Yoshihara, Asymptotic behavior of solutions of some difference equations defined by weakly dependent random vectors, Stoch. Anal. Appl., 33 (2015), 740-755.
    [5] H. Takahashi, T. Saigo, S. Kanagawa and K. Yoshihara, Optimal portfolios based on weakly dependent data, Dyn. Syst. Differ. Equ. Appl. , AIMS Proceedings, (2015), 1041-1049.
    [6] H.Takahashi, T. Saigo and K. Yoshihara, Approximation of optimal prices when basic data are weakly dependent, Dyn. Contin. Discrete Impuls. Syst. Ser. B, 23 (2016), 217-230.
    [7] K. Yoshihara, Asymptotic behavior of solutions of Black-Scholes type equations based on weakly dependent random variables, Yokohama Math. J., 58 (2012), 1-15.
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  • © 2017 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

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