Almost Sure Asymptotic Stability of the Euler-Maruyama Method With Random Variable Stepsizes for Stochastic Functional Differential Equations

MA Li; MA Ruinan

doi:10.21656/1000-0887.390057

Volume 40 Issue 1

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Article Navigation > Applied Mathematics and Mechanics > 2019 > 40(1): 97-107

MA Yong-qi, FENG Wei. Object-Oriented Finite Element Analysis and Programming in VC++[J]. Applied Mathematics and Mechanics, 2002, 23(12): 1283-1288.

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MA Li, MA Ruinan. Almost Sure Asymptotic Stability of the Euler-Maruyama Method With Random Variable Stepsizes for Stochastic Functional Differential Equations[J]. Applied Mathematics and Mechanics, 2019, 40(1): 97-107. doi: 10.21656/1000-0887.390057

MA Yong-qi, FENG Wei. Object-Oriented Finite Element Analysis and Programming in VC++[J]. Applied Mathematics and Mechanics, 2002, 23(12): 1283-1288.

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Almost Sure Asymptotic Stability of the Euler-Maruyama Method With Random Variable Stepsizes for Stochastic Functional Differential Equations

doi: 10.21656/1000-0887.390057

MA Li,
MA Ruinan

School of Mathematics and Statistics, Hainan Normal University, Haikou 571158, P.R.China

Funds: The National Natural Science Foundation of China(11861029)

Received Date: 2018-02-06
Rev Recd Date: 2018-08-22
Publish Date: 2019-01-01

Abstract

Abstract

The Euler-Maruyama (EM) approximation to a class of stochastic functional differential equations was studied. First, a numerical approximation with the EM method with random variable stepsizes was defined, then two characteristics of the random variable stepsizes were got: the summation of finite stepsizes is a stopping time and the summation of countably infinite stepsizes diverges. Finally, with the theory of non-negative semi-martingale convergence in discrete time, it was proved that the numerical approximation converges to zero almost surely if the coefficients satisfy the local Lipschitz condition and the monotonic condition. The results generalize the corresponding results of MAO Xuerong in a previous literature, where the EM approximation to a class of stochastic differential equations was studied and the numerical solution was proved to converge to zero almost surely.

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References

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