scholarly journals Convergence rate and stability of the split-step theta method for stochastic differential equations with piecewise continuous arguments

2019 ◽  
Vol 24 (2) ◽  
pp. 695-717
Author(s):  
Yulan Lu ◽  
◽  
Minghui Song ◽  
Mingzhu Liu
Keyword(s):  
Differential Equations ◽  
Convergence Rate ◽  
Stochastic Differential Equations ◽  
Piecewise Continuous ◽  
Piecewise Continuous Arguments ◽  
Theta Method

scholarly journals Strong convergence of the tamed Euler method for stochastic differential equations with piecewise continuous arguments and Poisson jumps

Filomat ◽  
2017 ◽  
Vol 31 (12) ◽  
pp. 3815-3836
Author(s):  
Huizi Yang ◽  
Minghui Song ◽  
Mingzhu Liu ◽  
Hong Wang
Keyword(s):  
Differential Equations ◽  
Convergence Rate ◽  
Stochastic Differential Equations ◽  
Polynomial Growth ◽  
Euler Method ◽  
Poisson Jumps ◽  
Lipschitz Continuous ◽  
Piecewise Continuous ◽  
Piecewise Continuous Arguments ◽  
Drift Coefficient

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.

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

2012 ◽  
Vol 2012 ◽  
pp. 1-21 ◽  
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.

Sign in / Sign up

Export Citation Format

Share Document