One-step approximations for stochastic functional differential equations

Buckwar, Evelyn

doi:10.1016/j.apnum.2005.05.001

Cited by 21 publications

(13 citation statements)

References 28 publications

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“…Moreover, because it is almost impossible to solve these equations explicitly, it is important to find some analytic and numerical approximations of the solutions. However, there is only a small amount of papers referring to such problems, Buckwar [3], Mao [14], for example. The present paper refers to an analytic method, which could lead to some constructions of appropriate numerical methods.…”

Section: Introduction and Preliminary Resultsmentioning

confidence: 97%

An approximate method via Taylor series for stochastic functional differential equations

Milošević

Jovanović

Janković

2010

Journal of Mathematical Analysis and Applications

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The subject of this paper is an analytic approximate method for stochastic functional differential equations whose coefficients are functionals, sufficiently smooth in the sense of Fréchet derivatives. The approximate equations are defined on equidistant partitions of the time interval, and their coefficients are general Taylor expansions of the coefficients of the initial equation. It will be shown that the approximate solutions converge in the L p -norm and with probability one to the solution of the initial equation, and also that the rate of convergence increases when degrees in Taylor expansions increase, analogously to real analysis.

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Section: Introduction and Preliminary Resultsmentioning

confidence: 97%

An approximate method via Taylor series for stochastic functional differential equations

Milošević

Jovanović

Janković

2010

Journal of Mathematical Analysis and Applications

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“…Otherwise, t n − τ (t n ) ≤ t 0 , Ȳ n := ψ(t n − τ (t n )) by (6). This is a trivial case, where by definition of Ȳ n we have E|…”

Section: Lemma 4 Assume Conditions (H2) and (I) (Ii) In Theorem 2 Hmentioning

confidence: 93%

“…In numerical analysis for stochastic differential equations (SDEs), convergence and stability are the two most important issues [7,8,[15][16][17]20,24,31,33,34]. For SDDEs, most of the existing works on numerical methods handle the cases which are of a constant lag τ and step size h being a fraction of τ [1,3,6,19,25,30]. However, DDEs with time-varying lag (1) and their stochastic counterpart (2) play an important role in engineering modeling.…”

Section: Introductionmentioning

confidence: 99%

θ-Maruyama methods for nonlinear stochastic differential delay equations

Wang

Gan

Wang

2015

Applied Numerical Mathematics

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Nonlinear stochastic differential delay equations Variable lag θ-Maruyama methods Strong convergence Exponential mean-square stability Mean-square stabilityIn this paper, mean-square convergence and mean-square stability of θ -Maruyama methods are studied for nonlinear stochastic differential delay equations (SDDEs) with variable lag. Under global Lipschitz conditions, the methods are proved to be mean-square convergent with order 1 2 , and exponential mean-square stability of SDDEs implies that of the methods for sufficiently small step size h > 0. Further, the exponential mean-square stability properties of SDDEs and those of numerical methods are investigated under some non-global Lipschitz conditions on the drift term. It is shown in this setting that the θ -Maruyama method with θ = 1 can preserve the exponential mean-square stability for any step size. Additionally, the θ -Maruyama method with 1 2 ≤ θ ≤ 1 is asymptotically mean-square stable for any step size, provided that the underlying system with constant lag is exponentially mean-square stable. Applications of this work to some special problem classes show that the results are deeper or sharper than those in the literature. Finally, numerical experiments are included to demonstrate the obtained theoretical results.

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“…Hence, it has received wide attention of researchers. The early related results can be found in Mao [1,2], Baker and Buckwar [3], Buckwar [4,5], Küchler and Platen [6], and the references therein. More recently, for the linear SDDEs, Cao et al [7], Liu et al [8], and Wang and Zhang [9] studied mean-square stability (MS-stability) of Euler-Maruyama, semi-implicit Euler-Maruyama, and Milstein methods, respectively.…”

Section: Introductionmentioning

confidence: 96%