Approximate solutions for a class of doubly perturbed stochastic differential equations

Mao, Wei; Hu, Liangjian; Mao, Xuerong

doi:10.1186/s13662-018-1490-5

Cited by 14 publications

(5 citation statements)

References 32 publications

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“…and the unique solution t x to the stochastic differential delay equation. [8] discussed the Caratheodory approximate solution for the class of doubly perturbed stochastic differential equation. [9] showed that stochastic differential equations with jumps and non-lipschitz coefficients have pair wise unique strong solutions by the Euler approximation method.…”

Section: Related Literaturementioning

confidence: 99%

A Comparative Survey of an Approximate Solution Method for Stochastic Delay Differential Equations

Emenonye¹,

Anonwa²

2023

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This study is focused on the approximate solution for the class of stochastic delay differential equations. The techniques applied involve the use of Caratheodory and Euler Maruyama procedures which approximated to stochastic delay differential equations. Based on the Caratheodory approximate procedure, it was proved that stochastic delay differential equations have unique solution and established that the Caratheodory approximate solution converges to the unique solution of stochastic delay differential equations under the Cauchy sequence and initial condition. This Caratheodory approximate procedure and Euler method both converge at the same rate. This is achieved by replacing the present state with past state. The existence and uniqueness of an approximate solution of the stochastic delay differential equation were shown and the approximate solution to the unique solution was also shown.

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Section: Related Literaturementioning

confidence: 99%

A Comparative Survey of an Approximate Solution Method for Stochastic Delay Differential Equations

Emenonye¹,

Anonwa²

2023

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“…However, most SDEs arising in practice are nonlinear, and cannot be solved explicitly. There has been tremendous interests in developing effective and reliable numerical methods for SDEs during the last few decades, for example see [4][5][6][7][8][9][10][11][12][13][14]. Runge-Kutta (RK) methods with continuous stage were firstly presented by Butcher in 1970s [15], and they have been investigated and discussed by several authors recently because of the great advantages in conserving symplecticity [16], preserving energy [17] and so on.…”

Section: Introductionmentioning

confidence: 99%

Continuous stage stochastic Runge–Kutta methods

2021

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In this work, a version of continuous stage stochastic Runge–Kutta (CSSRK) methods is developed for stochastic differential equations (SDEs). First, a general order theory of these methods is established by the theory of stochastic B-series and multicolored rooted tree. Then the proposed CSSRK methods are applied to three special kinds of SDEs and the corresponding order conditions are derived. In particular, for the single integrand SDEs and SDEs with additive noise, we construct some specific CSSRK methods of high order. Moreover, it is proved that with the help of different numerical quadrature formulas, CSSRK methods can generate corresponding stochastic Runge–Kutta (SRK) methods which have the same order. Thus, some efficient SRK methods are induced. Finally, some numerical experiments are presented to demonstrate those theoretical results.

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“…In the last decades, the Carathéodory approximation has been considered for various stochastic differential equations. Among others, we mention Turo [19] for stochastic functional differential equations, Mao [12,13] and Liu [11] for semilinear stochastic evolution equations with time delays, Ferrante & Rovira [8] for delay differential equations driven by fractional Brownian motion, Faizullah [7] for SDEs under G-Brownian motion, Benabdallah & Bourza [2] for perturbed SDEs with reflecting boundary, Mao et al [15] for doubly perturbed SDEs, etc.…”

Section: Introductionmentioning

confidence: 99%

Weak convergence of delay SDEs with applications to Carathéodory approximation

Son¹,

Dũng²,

Tan³

et al. 2021

Preprint

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In this paper, we consider a fundamental class of stochastic differential equations with time delays. Our aim is to investigate the weak convergence with respect to delay parameter of the solutions. Based on the techniques of Malliavin calculus, we obtain an explicit estimate for the rate of convergence. An application to the Carathéodory approximation scheme of stochastic differential equations is provided as well.

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Approximate solutions for a class of doubly perturbed stochastic differential equations

Cited by 14 publications

References 32 publications

A Comparative Survey of an Approximate Solution Method for Stochastic Delay Differential Equations

A Comparative Survey of an Approximate Solution Method for Stochastic Delay Differential Equations

Continuous stage stochastic Runge–Kutta methods

Weak convergence of delay SDEs with applications to Carathéodory approximation

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