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PRV property and the ϕ-asymptotic behavior of solutions of stochastic differential equations

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Abstract

In this paper, we investigate the a.s. asymptotic behavior of the solution of the stochastic differential equation dX(t) = g(X(t)) dt + σ(X(t))dW(t), X(0) ≢ 1, where g(·) and σ(·) are positive continuous functions, and W(·) is a standard Wiener process. By means of the theory of PRV functions we find conditions on g(·), σ(·), and ϕ(·) under which ϕ(X(·)) may be approximated a.s. by ϕ(μ(·)) on {X(t) → ∞}, where μ(·) is the solution of the ordinary differential equation dμ(t) = g(μ(t)) dt with μ(0) = 1.

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Authors and Affiliations

  1. Department of Mathematical Analysis and Probability Theory, National Technical University of Ukraine, pr. Peremogy, 37, Kyiv, 03056, Ukraine

    V. V. Buldygin & O. I. Klesov

  2. Mathematisches Institut, Universität zu Köln, Weyertal 86-90, D-50931, Köln, Germany

    J. G. Steinebach

Authors
  1. V. V. Buldygin
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  2. O. I. Klesov
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  3. J. G. Steinebach
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Correspondence to V. V. Buldygin.

Additional information

Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 4, pp. 445–465, October–December, 2007.

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Buldygin, V.V., Klesov, O.I. & Steinebach, J.G. PRV property and the ϕ-asymptotic behavior of solutions of stochastic differential equations. Lith Math J 47, 361–378 (2007). https://doi.org/10.1007/s10986-007-0025-7

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  • DOI: https://doi.org/10.1007/s10986-007-0025-7

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