Mean-square stability of the Euler–Maruyama method for stochastic differential delay equations with jumps

Tan, Jianguo; Wang, Hongli

doi:10.1080/00207160903474206

Cited by 17 publications

(8 citation statements)

References 16 publications

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“…Theorem 2. Suppose that the Lipschitz conditions and the growth conditions are satisfied, and the initial condition is in 2 (Ω, C, A 0 ). Then (1) subject to the initial condition (2) admits a unique solution.…”

Section: Preliminariesmentioning

confidence: 99%

“…Over the last couple of decades, a lot of work has been carried out to study differential equations with delay and/ or random noises, for example, [1][2][3][4][5]. The best-known and well-studied theory and systems include the delay differential equations (DDEs) presented by Kolmanvskii and Myshkis [6] and their stochastic generalizations and the stochastic delay differential equations (SDDEs) established by Mohammed [7,8], Mao [9,10], and Mohammed and Scheutzow [11].…”

Section: Introductionmentioning

confidence: 99%

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A Robust Weak Taylor Approximation Scheme for Solutions of Jump-Diffusion Stochastic Delay Differential Equations

Zhou¹,

et al. 2013

Abstract and Applied Analysis

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Stochastic delay differential equations with jumps have a wide range of applications, particularly, in mathematical finance. Solution of the underlying initial value problems is important for the understanding and control of many phenomena and systems in the real world. In this paper, we construct a robust Taylor approximation scheme and then examine the convergence of the method in a weak sense. A convergence theorem for the scheme is established and proved. Our analysis and numerical examples show that the proposed scheme of high order is effective and efficient for Monte Carlo simulations for jump-diffusion stochastic delay differential equations.

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Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Robust Weak Taylor Approximation Scheme for Solutions of Jump-Diffusion Stochastic Delay Differential Equations

Zhou¹,

et al. 2013

Abstract and Applied Analysis

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“…For example, Wang et al [11] constructed the semiimplicit Euler method for stochastic differential delay equation with jumps. Tan and Wang [5] investigated the mean-square stability of the explicit Euler method for linear SDDEs with jumps. Zhang et al [8] derived some criteria on pth moment stability and almost sure stability with general decay rates of stochastic differential delay equations with Poisson jumps and Markovian switching.…”

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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“…Stochastic θ -method includes the commonly used EM method and backward Euler-Maruyama (BEM) method by choosing θ = 0 and θ = 1, and it is more general than these two methods (Cao, Liu, & Fan, 2004;Higham & Kloeden, 2005;Higham, Mao, & Yuan, 2007;Kloeden & Platen, 1992;Li & Cao, 2015;Li & Gan, 2011;Tan & Wang, 2011;Wang, Mei, & Xue, 2007;Wu, Mao, & Szpruch, 2010). In recent decades, stochastic θ -method has been increasingly used to cope with NSDDEs and stochastic delay differential equations (SDDEs) with jumps.…”

Section: Introductionmentioning

confidence: 99%

Exponential mean-square stability of the θ-method for neutral stochastic delay differential equations with jumps

Zhao

Deng

2016

International Journal of Systems Science

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KEYWORDSNeutral stochastic delay differential equations with jumps; stochastic θ -method; exponential mean-square stability ABSTRACTThe exponential mean-square stability of the θ-method for neutral stochastic delay differential equations (NSDDEs) with jumps is considered. With some monotone conditions, the trivial solution of the equation is proved to be exponentially mean-square stable. If the drift coefficient and the parameters satisfy more strengthened conditions, for the constrained stepsize, it is shown that the θ-method can preserve the exponential mean-square stability of the trivial solution for θ ࢠ [0, 1]. Since θ-method covers the commonly used Euler-Maruyama (EM) method and the backward Euler-Maruyama (BEM) method, the results are valid for the above two methods. Moreover, they can adapt to the NSDDEs and the stochastic delay differential equations (SDDEs) with jumps. Finally, a numerical example illustrates the effectiveness of the theoretical results.

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Mean-square stability of the Euler–Maruyama method for stochastic differential delay equations with jumps

Cited by 17 publications

References 16 publications

A Robust Weak Taylor Approximation Scheme for Solutions of Jump-Diffusion Stochastic Delay Differential Equations

A Robust Weak Taylor Approximation Scheme for Solutions of Jump-Diffusion Stochastic Delay Differential Equations

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

Exponential mean-square stability of the θ-method for neutral stochastic delay differential equations with jumps

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