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Partial asymptotic stability in probability of stochastic differential equations

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  • Ignatyev, Oleksiy

Abstract

A system of stochastic differential equations which has a zero solution X=0 is considered. It is assumed that there exists a function V(t,x), positive definite with respect to part of the state variables which also has the infinitesimal upper limit with respect to the part of the variables and such that the corresponding operator LV is nonpositive. It is proved that if the nondegeneracy condition of the matrix corresponding to the coefficients of Wiener processes holds with respect to the part of the variables with appropriate function r(x), then the zero solution is asymptotically stable in probability with respect to the part of the variables.

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  • Ignatyev, Oleksiy, 2009. "Partial asymptotic stability in probability of stochastic differential equations," Statistics & Probability Letters, Elsevier, vol. 79(5), pages 597-601, March.
    • Handle: RePEc:eee:stapro:v:79:y:2009:i:5:p:597-601
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    Cited by:

    1. Sakthivel, R. & Luo, J., 2009. "Asymptotic stability of nonlinear impulsive stochastic differential equations," Statistics & Probability Letters, Elsevier, vol. 79(9), pages 1219-1223, May.
    2. Socha, Leslaw & Zhu, Quanxin, 2019. "Exponential stability with respect to part of the variables for a class of nonlinear stochastic systems with Markovian switchings," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 155(C), pages 2-14.

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