PRV property of functions and the asymptotic behaviour of solutions of stochastic differential equations

Buldygin, V.; Klesov, O.; Steinebach, J.

doi:10.1090/S0094-9000-06-00660-0

PRV property of functions and the asymptotic behaviour of solutions of stochastic differential equations

Authors: V. V. Buldygin, O. I. Klesov and J. G. Steinebach
Translated by: The authors
Journal: Theor. Probability and Math. Statist. 72 (2006), 11-25
MSC (2000): Primary 60H10; Secondary 34D05, 60F15, 60G17
DOI: https://doi.org/10.1090/S0094-9000-06-00660-0
Published electronically: August 10, 2006
MathSciNet review: 2168132
Full-text PDF Free Access

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Abstract: In this paper, we investigate the a.s. asymptotic behaviour of the solution of the stochastic differential equation $dX(t) = g(X(t)) dt + \sigma (X(t)) dW(t)$, where $g(\boldsymbol \cdot )$ and $\sigma (\boldsymbol \cdot )$ are positive continuous functions and $W(\boldsymbol \cdot )$ is a standard Wiener process. By an application of the theory of PRV and PMPV functions, we find conditions on $g(\boldsymbol \cdot )$ and $\sigma (\boldsymbol \cdot )$, under which $X(\boldsymbol \cdot )$ may be approximated a.s. on $\{X(t)\to \infty \}$ by the solution of the deterministic differential equation $d\mu (t) = g(\mu (t)) dt$. Moreover, we study the asymptotic stability with respect to initial conditions of solutions of the above SDE as well as the asymptotic behaviour of generalized renewal processes connected with this SDE.

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