PRV property and the <Emphasis Type="Italic">ϕ</Emphasis>-asymptotic behavior of solutions of stochastic differential equations |
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Authors: | V V Buldygin O I Klesov J G Steinebach |
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Institution: | 1.Department of Mathematical Analysis and Probability Theory,National Technical University of Ukraine,Kyiv,Ukraine;2.Mathematisches Institut,Universit?t zu K?ln,K?ln,Germany |
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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.
Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 4, pp. 445–465, October–December, 2007. |
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Keywords: | stochastic differential equation asymptotic behavior of solutions pseudo-regulary varying function |
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