Strong convergence and exponential stability of stochastic differential equations with piecewise continuous arguments for non-globally Lipschitz continuous coefficients

In the present work, the tamed Euler method is proven to be strongly convergent for stochastic differential equations with piecewise continuous arguments and Poisson jumps, where the diffusion and jump coefficients are globally Lipschitz continuous, the drift coefficient is one-sided Lipschitz continuous, and its derivative demonstrates an at most polynomial growth. Moreover, the convergence rate is obtained.

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Convergence and stability of the one-leg θ method for stochastic differential equations with piecewise continuous arguments

Filomat ◽

10.2298/fil1903945l ◽

2019 ◽

Vol 33 (3) ◽

pp. 945-960

Author(s):

Yulan Lu ◽

Minghui Song ◽

Mingzhu Liu

Keyword(s):

Differential Equations ◽

Exponential Stability ◽

Stochastic Differential Equations ◽

Step Size ◽

Almost Sure Exponential Stability ◽

Exponentially Stable ◽

Piecewise Continuous ◽

Piecewise Continuous Arguments ◽

The Stability ◽

The One

The equivalent relation is established here about the stability of stochastic differential equations with piecewise continuous arguments(SDEPCAs) and that of the one-leg ? method applied to the SDEPCAs. Firstly, the convergence of the one-leg ? method to SDEPCAs under the global Lipschitz condition is proved. Secondly, it is proved that the SDEPCAs are pth(p 2 (0; 1)) moment exponentially stable if and only if the one-leg ? method is pth moment exponentially stable for some sufficiently small step-size. Thirdly, the corollaries that the pth moment exponential stability of the SDEPCAs (the one-leg ? method) implies the almost sure exponential stability of the SDEPCAs (the one-leg ? method) are given. Finally, numerical simulations are provided to illustrate the theoretical results.

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Numerical Solutions of Stochastic Differential Equations with Piecewise Continuous Arguments under Khasminskii-Type Conditions

Journal of Applied Mathematics ◽

10.1155/2012/696849 ◽

2012 ◽

Vol 2012 ◽

pp. 1-21 ◽

Cited By ~ 3

Author(s):

Minghui Song ◽

Ling Zhang

Keyword(s):

Differential Equations ◽

Stochastic Differential Equations ◽

Growth Condition ◽

Linear Growth ◽

Numerical Solutions ◽

Type Theorem ◽

Global Existence Of Solutions ◽

Linear Growth Condition ◽

Piecewise Continuous ◽

Piecewise Continuous Arguments

The main purpose of this paper is to investigate the convergence of the Euler method to stochastic differential equations with piecewise continuous arguments (SEPCAs). The classical Khasminskii-type theorem gives a powerful tool to examine the global existence of solutions for stochastic differential equations (SDEs) without the linear growth condition by the use of the Lyapunov functions. However, there is no such result for SEPCAs. Firstly, this paper shows SEPCAs which have nonexplosion global solutions under local Lipschitz condition without the linear growth condition. Then the convergence in probability of numerical solutions to SEPCAs under the same conditions is established. Finally, an example is provided to illustrate our theory.

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