Convergence and stability of stochastic theta method for nonlinear stochastic differential equations with piecewise continuous arguments

2021 ◽  
pp. 113849
Author(s):  
Yuhang Zhang ◽  
Minghui Song ◽  
Mingzhu Liu
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Nonlinear Stochastic Differential Equations ◽  
Convergence And Stability ◽  
Piecewise Continuous ◽  
Piecewise Continuous Arguments ◽  
Stochastic Theta Method ◽  
Theta Method

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.

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