Strong convergence of the partially truncated Euler–Maruyama method for a class of stochastic differential delay equations

Zhang, Wei; Song, Meimei; Liu, M.Z.

doi:10.1016/j.cam.2017.11.030

Cited by 38 publications

(18 citation statements)

References 10 publications

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“…Since f does not satisfy polynomial growth condition in this case, then the strong convergence result Theorem 3.7 in [18] does not be hold here. However, for the continuous-time MTEM methods (2.6) and (2.7), the strong convergence results still holds for the given NSDDE.…”

Section: Examplesmentioning

confidence: 97%

“…Thus numerical methods for NSDDEs (1.1) have been playing more and more important roles.The convergence of the numerical methods for NSDDEs (1.1) have been discussed intensively by many researchers, for example, Gan et. al [4] investigated mean square convergence of stochastic θ method under global Lipschitz condition, [17] studied Mean square convergence of one-step methods under the same assumptions, [13] considered convergence in probability of the backward Euler approximate solution for a class of stochastic differential equations with constant delay, Zhang et.al considered strong convergence of the partially truncated Euler-Maruyama method for a class of stochastic differential delay equations in [18], there are also many other literatures concerning with this topic, see e.g. [2,6,11,12,19,20].Recently, in [9], Mao developed a new explicit numerical simulation method, called truncated EM method.…”

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confidence: 99%

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Strong convergence rates of modified truncated EM methods for neutral stochastic differential delay equations

Lan

Wang

2019

Journal of Computational and Applied Mathematics

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The aim of this paper is to investigate strong convergence of modified truncated Euler-Maruyama method for neutral stochastic differential delay equations introduced in Lan (2018). Strong convergence rates of the given numerical scheme to the exact solutions at fixed time T are obtained under local Lipschitz and Khasminskii-type conditions. Moreover, convergence rates over a time interval [0, T ] are also obtained under additional polynomial growth condition on g without the weak monotonicity condition (which is usually the standard assumption to obtain the convergence rate). Two examples are presented to interpret our conclusions.MSC 2010: 60H10, 65C30, 65L20.Recently, such neutral stochastic differential delay equations (short for NSDDEs) have been found more and more applications in many fields such as control theory, electrodynamics, biomathematics and so on. However, most NSDDEs can not be solved explicitly except some special ones. Thus numerical methods for NSDDEs (1.1) have been playing more and more important roles.The convergence of the numerical methods for NSDDEs (1.1) have been discussed intensively by many researchers, for example, Gan et. al [4] investigated mean square convergence of stochastic θ method under global Lipschitz condition, [17] studied Mean square convergence of one-step methods under the same assumptions, [13] considered convergence in probability of the backward Euler approximate solution for a class of stochastic differential equations with constant delay, Zhang et.al considered strong convergence of the partially truncated Euler-Maruyama method for a class of stochastic differential delay equations in [18], there are also many other literatures concerning with this topic, see e.g. [2,6,11,12,19,20].Recently, in [9], Mao developed a new explicit numerical simulation method, called truncated EM method. Strong convergence theory were established there under local Lipschitz condition plus the Khasminskii-type condition. And then he obtained sufficient conditions for the strong convergence rate of it in [10]. Motivated by these two works, Lan and Xia introduced in [8] modified truncated Euler-Maruyama (MTEM) method and obtained the strong convergence rate under given conditions. Then in [7], the author generalized the MTEM method from SDEs cases to the NSDDEs cases and obtain asymptotic exponential stability of it under given conditions. However, the strong convergence of the MTEM method is still not known, which is the main topic of this paper.Although strong convergence of the given numerical methods are considered in many papers such as [1,2,4,9,16,17] and so on, the convergence rates are not known. Some other papers considered strong convergence rates of the given numerical methods under weak monotonicity condition. For example, in Guo et al [3], Assumption 5.1 is necessary to obtain the convergence rate of the truncated EM method at fixed time T , in Tan and Yuan [14], A8 is needed to obtain the convergence rate of the truncated EM method over a time interval [0, T ], for the wea...

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Section: Examplesmentioning

confidence: 97%

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confidence: 99%

Strong convergence rates of modified truncated EM methods for neutral stochastic differential delay equations

Lan

Wang

2019

Journal of Computational and Applied Mathematics

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“…where diffusion function P(u) and advection function Q(u) make equation (3.8) nonlinear. Equation (3.8) and its special cases are used to describe many particular phenomena, such as anomalous diffusion, random walkers in an expanding medium and transport dynamics in complex systems [3,4], and are studied with analytical and numerical methods (see [29,30] and references therein). For example, the linear fractional-type advection-diffusion equation is considered using the Lie symmetry method [29].…”

Section: (B) the Time-fractional Diffusion-convection Equationmentioning

confidence: 99%

Symmetry determination and nonlinearization of a nonlinear time-fractional partial differential equation

Zhang

2020

Proc. R. Soc. A.

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We first show that the infinitesimal generator of Lie symmetry of a time-fractional partial differential equation (PDE) takes a unified and simple form, and then separate the Lie symmetry condition into two distinct parts, where one is a linear time-fractional PDE and the other is an integer-order PDE that dominates the leading position, even completely determining the symmetry for a particular type of time-fractional PDE. Moreover, we show that a linear time-fractional PDE always admits an infinite-dimensional Lie algebra of an infinitesimal generator, just as the case for a linear PDE and a nonlinear time-fractional PDE admits, at most, finite-dimensional Lie algebra. Thus, there exists no invertible mapping that converts a nonlinear time-fractional PDE to a linear one. We illustrate the results by considering two examples.

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“…Example 1. We consider the following example [21] dy(t) = [−2y(t) + y(t − 1) − y 5 (t)]dt + y 2 (t)dW(t),…”

Section: Numerical Experimentsmentioning

confidence: 99%

Projected Euler-Maruyama method for stochastic delay differential equations under a global monotonicity condition

Huang

2020

Applied Mathematics and Computation

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Strong convergence of the partially truncated Euler–Maruyama method for a class of stochastic differential delay equations

Cited by 38 publications

References 10 publications

Strong convergence rates of modified truncated EM methods for neutral stochastic differential delay equations

Strong convergence rates of modified truncated EM methods for neutral stochastic differential delay equations

Symmetry determination and nonlinearization of a nonlinear time-fractional partial differential equation

Projected Euler-Maruyama method for stochastic delay differential equations under a global monotonicity condition

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