Strong convergence of the split-step theta method for neutral stochastic delay differential equations

2017 ◽  
Vol 120 ◽  
pp. 215-232 ◽  
Author(s):  
Zhiping Yan ◽  
Aiguo Xiao ◽  
Xiao Tang
Keyword(s):  
Differential Equations ◽  
Strong Convergence ◽  
Delay Differential Equations ◽  
Stochastic Delay Differential Equations ◽  
Stochastic Delay ◽  
Delay Differential ◽  
Theta Method

scholarly journals Strong Convergence of the Split-Step Theta Method for Stochastic Delay Differential Equations with Nonglobally Lipschitz Continuous Coefficients

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Chao Yue ◽  
Chengming Huang
Keyword(s):  
Differential Equations ◽  
Strong Convergence ◽  
Delay Differential Equations ◽  
Stochastic Delay Differential Equations ◽  
Stochastic Delay ◽  
Lipschitz Continuous ◽  
Highly Nonlinear ◽  
Nonautonomous Equations ◽  
Delay Differential ◽  
Theta Method

This paper is concerned with the convergence analysis of numerical methods for stochastic delay differential equations. We consider the split-step theta method for nonlinear nonautonomous equations and prove the strong convergence of the numerical solution under a local Lipschitz condition and a coupled condition on the drift and diffusion coefficients. In particular, these conditions admit that the diffusion coefficient is highly nonlinear. Furthermore, the obtained results are supported by numerical experiments.

scholarly journals On Mean Square Stability and Dissipativity of Split-Step Theta Method for Nonlinear Neutral Stochastic Delay Differential Equations

2016 ◽  
Vol 2016 ◽  
pp. 1-8
Author(s):  
Haiyan Yuan ◽  
Jihong Shen ◽  
Cheng Song
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Mean Square ◽  
Stochastic Delay Differential Equations ◽  
Stochastic Delay ◽  
Mean Square Stability ◽  
Linear Growth Condition ◽  
The Mean ◽  
Delay Differential ◽  
Theta Method

A split-step theta (SST) method is introduced and used to solve the nonlinear neutral stochastic delay differential equations (NSDDEs). The mean square asymptotic stability of the split-step theta (SST) method for nonlinear neutral stochastic delay differential equations is studied. It is proved that under the one-sided Lipschitz condition and the linear growth condition, the split-step theta method withθ∈(1/2,1]is asymptotically mean square stable for all positive step sizes, and the split-step theta method withθ∈[0,1/2]is asymptotically mean square stable for some step sizes. It is also proved in this paper that the split-step theta (SST) method possesses a bounded absorbing set which is independent of initial data, and the mean square dissipativity of this method is also proved.

scholarly journals A Robust Weak Taylor Approximation Scheme for Solutions of Jump-Diffusion Stochastic Delay Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Yanli Zhou ◽  
Yonghong Wu ◽  
Xiangyu Ge ◽  
B. Wiwatanapataphee
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Approximation Scheme ◽  
Jump Diffusion ◽  
Mathematical Finance ◽  
Stochastic Delay Differential Equations ◽  
Stochastic Delay ◽  
Taylor Approximation ◽  
Wide Range ◽  
Delay Differential

Stochastic delay differential equations with jumps have a wide range of applications, particularly, in mathematical finance. Solution of the underlying initial value problems is important for the understanding and control of many phenomena and systems in the real world. In this paper, we construct a robust Taylor approximation scheme and then examine the convergence of the method in a weak sense. A convergence theorem for the scheme is established and proved. Our analysis and numerical examples show that the proposed scheme of high order is effective and efficient for Monte Carlo simulations for jump-diffusion stochastic delay differential equations.

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