Mean square stability of the solutions of autonomous dynamic diffusion systems with finite aftereffect with regard for r
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MEAN SQUARE STABILITY OF THE SOLUTIONS OF AUTONOMOUS DYNAMIC DIFFUSION SYSTEMS WITH FINITE AFTEREFFECT WITH REGARD FOR RANDOM FACTORS V. K. Yasinskya† and N. P. Bodrycka‡
UDC 519.21
Abstract. The necessary and sufficient conditions are obtained for the asymptotic mean square stability of strong solutions of autonomous diffusion stochastic functional-differential equations with finite aftereffect and random factors (random functions with different distribution) taken into account. Keywords: mean square stability, diffusion stochastic functional differential equation, Cauchy problem, fundamental solution, random factor. INTRODUCTION Stochastic differential equations without aftereffect with respect to the existence of a strong solution, their properties and asymptotic behavior were considered by I. I. Gikhman and A. V. Skorokhod in their fundamental works [1–5], where references to similar studies of foreign authors can be found. Stochastic functional differential systems are analyzed in the well-known monographs by V. B. Kolmanovskii, L. E. Shaikhet, and R. Z. Khas’minskii [6–8]. The fundamental issues of the asymptotic behavior of the solutions of stochastic functional differential equations (SFDEs) and of their applications are presented at a high fundamental level in the popular monograph by E. F. Tsar’kov [9]. In the present paper, we will generalize some Tsar’kov’s results under the presence of random factors in SFDE. We will pass to the integral notation of the solution of a diffusion stochastic functional differential equation (DSFDE) using the fundamental solution of the respective deterministic functional differential equation (FDE). We will obtain the necessary and sufficient conditions for the asymptotic mean square stability (l.i.m.) of the strong solution of an autonomous DSFDE with finite aftereffect and random factors (external action of random functions with different distributions) taken into account. We will analyze the behavior of strong solutions of nonautonomous DSFDEs at infinity in view of random factors and consider the case of a linear nonautonomous DSFDE of a more general form. The results will be illustrated by model problems, where the influence of random functions on the mathematical model of the system under study is taken into account. REPRESENTING THE SOLUTION OF THE DSFDE BY THE FUNDAMENTAL SOLUTION OF THE RESPECTIVE DETERMINISTIC FDE Let a linear DSFDE with random factors dy( t ) = a( y t )dt + f ( x ( w ))b ( y t )dW ( t )
(1)
y( q + s ) = j( q ) for
(2)
with the initial conditions " q Î [ -t , 0], t > 0,
a
Yu. Fed’kovych National University, Chernivtsi, Ukraine, †[email protected]; ‡[email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 3, May–June, 2012, pp. 127–141. Original article submitted July 21, 2011. 1060-0396/12/4803-0429
©
2012 Springer Science+Business Media, Inc.
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be given on a probability space (W , F , P ) with filtration {Ft , t ³ 0}, Ft Ì F, where a( j ) is a continuous linear mapping from C n ([ -t , 0]) into R n ; b( j ) is a conti
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