Abstract
In this paper, we concern stability of numerical methods applied to stochastic delay integro-differential equations. For linear stochastic delay integro-differential equations, it is shown that the mean-square stability is derived by the split-step backward Euler method without any restriction on step-size, while the Euler–Maruyama method could reproduce the mean-square stability under a step-size constraint. We also confirm the mean-square stability of the split-step backward Euler method for nonlinear stochastic delay integro-differential equations. The numerical experiments further verify the theoretical results.
Highlights
Stochastic delay integro-differential equations, as the mathematical model, widely apply in biology, physics, economics and finance [1, 2]
Because of the stochastic delay integrodifferential equations themselves, it is not easy to obtain an explicit solution for these kinds of equations, so it is necessary to research the numerical methods for numerical solution of stochastic delay integro-differential equations [3, 4]
Ding et al [5] dealt with the stability of the semi-implicit Euler method for linear stochastic delay integro-differential equations
Summary
Stochastic delay integro-differential equations, as the mathematical model, widely apply in biology, physics, economics and finance [1, 2]. There are few results on the numerical methods to stochastic delay integro-differential equations. Ding et al [5] dealt with the stability of the semi-implicit Euler method for linear stochastic delay integro-differential equations. Rathinasamy and Balachandran [6] proved mean-square stability of the Milstein method for linear stochastic delay integrodifferential equations with Markovian switching under suitable conditions on the integral term. Rathinasamy and Balachandran [9] analyzed T-stability of the split-step-θ -methods for linear stochastic delay integro-differential equations. Wu [10] investigated the mean-square stability for stochastic delay integro-differential equations by the strong balanced methods and the weak balanced methods with sufficiently small step-size. Zhang and Li Journal of Inequalities and Applications (2018) 2018:114 method for stochastic delay integro-differential equations in Sect.
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