Strong Law of Large Numbers for Solutions of Non-Autonomous Stochastic Differential Equations

Authors

DOI:

https://doi.org/10.20535/1810-0546.2017.4.106506

Keywords:

Strong law of large numbers, Stochastic differential equation, Wiener process, Asymptotic behavior

Abstract

Background. Asymptotic behavior at infinity of non-autonomous stochastic differential equation solutions is studied in the paper.

 Objective. The aim of the work is to find sufficient conditions for the strong law of large numbers for a random process which is a solution of non-autonomous stochastic differential equation.

Methods. Basic results of the theory of stochastic differential equations related to stochastic integrals estimation.

Results. Sufficient conditions for almost sure convergence to zero of normalized term related to diffusion of non-autonomous stochastic differential equation are obtained.

Conclusions. Results of the paper can be used for further research on the asymptotic behavior of stochastic differential equation solutions, finding the stability condition of stochastic differential equation solution and ergodic type problems also.

References

I.I. Gihman and A.V. Skorohod, Stochastic Differential Equations and its Applications. Kyiv, SU: Naukova Dumka, 1982 (in Russian).

J.A.D. Appleby et al., “On the asymptotic stability of a class of perturbed ordinary differential equations with weak asymptotic mean reversion”, E. J. Qualitative Theory of Diff. Equ., Proc. 9th Coll., vol. 1, pp. 1–36, 2011.

J.A.D. Appleby et al., “Characterisation of the asymptotic behaviour of scalar linear differential equations with respect to a fading stochastic perturbation”, Discrete Contin. Dyn. Syst., Suppl., vol. 2011, pp. 79–90, 2011.

V.V. Buldygin et al., Pseudo Regularly Varying Functions and Generalized Renewal Processes. Kyiv, Ukraine: TBiMC, 2012 (in Ukrainian).

O.A. Tymoshenko, “Generalization of asymptotic behavior of nonautonomouse stochastic differential equations”, Naukovi Visti NTUU KPI, no. 4, pp. 100–106, 2016. doi: 10.20535/1810-0546.2016.4.71649

O.I. Klesov and O.A. Tymoshenko, “Unbounded solutions of stochastic differential equations with time-dependent coeffi­ci­ents”, Annales Univ. Sci. Budapest., Sect. Comp., vol. 41, pp. 25–35, 2013.

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Published

2017-09-18

Issue

No. 4 (2017): Physics and Mathematics

Section

Art

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Copyright (c) 2017 Igor Sikorsky Kyiv Polytechnic Institute

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