Pathwise Stability of Degenerate Stochastic Evolutions

Bierkens, Joris

doi:10.1007/s00020-010-1841-4

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Dy(t) = αY(t)dt + βY(t − R )Dt + σ Y(t)dw(t)1

Stability1

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Cited by 2 publications

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“…(1.6) could secure the same exponential stability in the almost sure sense as long as the delay parameter r > 0 is sufficiently small. In fact, it is shown in Bierkens [3] that if r > 0 is so small that |β|e 3σ 2 r/2 + α < σ 2 /2, then the null solution of (1.6) is exponentially stable almost surely. Thus, stability is not sensitive in this case to small delays.…”

Section: Dy(t) = αY(t)dt + βY(t − R )Dt + σ Y(t)dw(t)mentioning

confidence: 99%

Sensitivity to Small Delays of Pathwise Stability for Stochastic Retarded Evolution Equations

2017

J Theor Probab

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In this paper, we shall study the almost sure pathwise exponential stability property for a class of stochastic functional differential equations with delays, possibly, in the highest-order derivative terms driven by multiplicative noise. Instead of establishing a moment exponential stability as the first step and then proceeding to investigate the pathwise stability of the system under consideration, we shall develop a direct approach for this problem. As a consequence, we can show that some systems, which are not exponential momently stable, have the exponential stability not sensitive to small delays in the almost sure sense.

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Section: Dy(t) = αY(t)dt + βY(t − R )Dt + σ Y(t)dw(t)mentioning

confidence: 99%

Sensitivity to Small Delays of Pathwise Stability for Stochastic Retarded Evolution Equations

2017

J Theor Probab

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show abstract

“…It is also possible to obtain the following stability result using the dissipativity property of a delay semigroup (see [2], Section 6.6.3, and [3]).…”

Section: Stabilitymentioning

confidence: 99%