Exponential stability in p-th mean of solutions, and of convergent Euler-type solutions, of stochastic delay differential equations

Baker, Christopher T. H.; Buckwar, Evelyn

doi:10.1016/j.cam.2005.01.018

Cited by 181 publications

(101 citation statements)

References 15 publications

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“…With the aid of a Lyapunov-type theorem we obtain sufficient conditions on the parameters in (2.3) to ensure asymptotic mean-square stability of the zero solution of the recurrence equation. Related methods have been used in the case of stochastic delay differential equations in [2]. In [15] a general method is described to construct appropriate Lyapunov functionals and in fact, it is possible to construct different Lyapunov functionals giving different sets of sufficient conditions.…”

Section: Lyapunov Functionals For Stochastic Difference Equationsmentioning

confidence: 99%

Asymptotic Mean-Square Stability of Two-Step Methods for Stochastic Ordinary Differential Equations

2006

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Abstract.We deal with linear multi-step methods for SDEs and study when the numerical approximation shares asymptotic properties in the mean-square sense of the exact solution.As in deterministic numerical analysis we use a linear time-invariant test equation and perform a linear stability analysis. Standard approaches used either to analyse deterministic multi-step methods or stochastic one-step methods do not carry over to stochastic multi-step schemes. In order to obtain sufficient conditions for asymptotic mean-square stability of stochastic linear two-step-Maruyama methods we construct and apply Lyapunov-type functionals. In particular we study the asymptotic meansquare stability of stochastic counterparts of two-step Adams-Bashforth-and AdamsMoulton-methods, the Milne-Simpson method and the BDF method.

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Section: Lyapunov Functionals For Stochastic Difference Equationsmentioning

confidence: 99%

Asymptotic Mean-Square Stability of Two-Step Methods for Stochastic Ordinary Differential Equations

2006

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“…Since most of these equations cannot be solved explicitly, numerical approximations become an important tool in studying the properties of these stochastic systems (see [1], [3], [10], [17], [20] and [26]). Since stability is one of the major concerns in systems analysis, the stability of numerical methods becomes also one of the main tools to examine the stability of the exact solution of the stochastic systems (see [18], [21] and [24]).…”

Section: Introductionmentioning

confidence: 99%

A Note on the Almost Sure Exponential Stability of the Milstein Method for Stochastic Delay Differential Equations With Jumps

Rouz¹,

Ahmadian²,

Akbarfam³

et al. 2017

Int. J. of Pure and Appl. Math.

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In this paper we consider the Milstein method to investigate the almost sure exponential stability for nonlinear stochastic delay differential equations (SDDEs) with jump diffusion, that is, processes that change only by jumps. The class of jump-diffusion SDDEs that admits explicit solutions is rather limited. In this regard, we employ the continuous and discrete semimartingale convergence theorem. It is shown that the Milstein method reproduces the almost sure exponential stability under an additional linear growth condition on drift and the function L 1 g. In the numerical section we present the results by simulating the method on kind of equations.

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“…For delay differential equations or stochastic differential equations, stability conditions have been well investigated. However, for stochastic delay differential equations, only a few conditions were derived for exponential p-th moment stability (see [3]- [5] for example). In [6], almost sure stability was studied for a second order stochastic differential equation (with no delay) which is a model of a hinged supported inverted pendulum and showed that noise can enhance stability.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic Mean Square Stability Analysis for a Stochastic Delay Differential Equation

Xue

Yamamoto

2012

SICE Journal of Control, Measurement, and System Integration

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: For a stochastic delay differential equation, a stability condition is derived in the sense of mean square stability. The condition substantiates the fact that noise with an appropriate power can reduce the influence of time delay in the differential equations. It is also illustrated by a numerical example. To derive the stability condition, the so-called domain subdivision method and Ito's formula are adopted.

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Exponential stability in p-th mean of solutions, and of convergent Euler-type solutions, of stochastic delay differential equations

Abstract: Exponential stability in p-th mean of solutions, and of convergent Euler-type solutions, of stochastic delay differential equations

Cited by 181 publications

References 15 publications

Asymptotic Mean-Square Stability of Two-Step Methods for Stochastic Ordinary Differential Equations

Asymptotic Mean-Square Stability of Two-Step Methods for Stochastic Ordinary Differential Equations

A Note on the Almost Sure Exponential Stability of the Milstein Method for Stochastic Delay Differential Equations With Jumps

Asymptotic Mean Square Stability Analysis for a Stochastic Delay Differential Equation

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