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Theory of Probability and Mathematical Statistics

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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
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 72 (2005).
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
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Abstract | References | Similar Articles | Additional Information

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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Additional Information

V. V. Buldygin
Affiliation: Department of Mathematical Analysis and Probability Theory, National Technical University of Ukraine (KPI), Pr. Peremogy 37, Kyiv 03056, Ukraine
Email: valbuld@comsys.ntu-kpi.kiev.ua

O. I. Klesov
Affiliation: Department of Mathematical Analysis and Probability Theory, National Technical University of Ukraine (KPI), Pr. Peremogy 37, Kyiv 03056, Ukraine
Email: oleg@tbimc.freenet.kiev.ua

J. G. Steinebach
Affiliation: Mathematisches Institut, Universität zu Köln, Weyertal 86–90, D–50931 Köln, Germany
Email: jost@math.uni-koeln.de

DOI: https://doi.org/10.1090/S0094-9000-06-00660-0
Received by editor(s): July 15, 2004
Published electronically: August 10, 2006
Additional Notes: This work has partially been supported by Deutsche Forschungsgemeinschaft under DFG grants 436 UKR 113/41/0-2 and 436 UKR 113/68/0-1.
Article copyright: © Copyright 2006 American Mathematical Society

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