Averaging method and two-sided bounded solutions of Itô stochastic systems

Самойленко, А. М.; Makhmudov, N. I.; Stanzhitskii, A. N.

doi:10.1134/s0012266107010089

Cited by 4 publications

(3 citation statements)

References 3 publications

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“…Borysenko ([13] - [15]). The averaging method on the infinite interval for the system of Ito stochastic differential equations is studied in [16].…”

Section: Introductionmentioning

confidence: 99%

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Non-Autonomous Random Oscillating Systems of Fourth Order under Small Periodical External Perturbations with Jumps

Borysenko¹,

Borysenko

2019

Stat., optim. inf. comput.

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The asymptotic behavior of a non-autonomous oscillating system described by a differential equation of the fourth order with small non-linear periodical external perturbations of "white noise", non-centered and centered "Poisson noise" types is studied. Each term of external perturbations has own order of a small parameter ε. If the small parameter is equal to zero, then the general solution of the obtained non-stochastic fourth order differential equation has an oscillating part. We consider the given differential equation with external stochastic perturbations as the system of stochastic differential equations and study the limit behavior of its solution at the time moment t/ε k , as ε → 0. The system of averaging stochastic differential equations is derived and its dependence on the order of the small parameter in each term of external perturbations is studied. The non-resonance and resonance cases are considered.

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“…Borysenko ([13] - [15]). The averaging method on the infinite interval for the system of Ito stochastic differential equations is studied in [16].…”

Section: Introductionmentioning

confidence: 99%

“…Since A εm (t) →Ā(t), ζ εm (t) →ζ(t) in probability, as ε m → 0, then, using (17), from (15) and (16) we obtain estimates…”

mentioning

confidence: 99%

Non-Autonomous Random Oscillating Systems of Fourth Order under Small Periodical External Perturbations with Jumps

Borysenko¹,

Borysenko

2019

Stat., optim. inf. comput.

View full text Add to dashboard Cite

show abstract

“…The stochastic case was studied mainly for equations driven by Wiener process. Averaging of equations was considered in [18], slow-fast systems -in [2], [4, Section 7.9], [25].…”

Section: Introductionmentioning

confidence: 99%