scholarly journals Numerical Analysis for Stochastic Partial Differential Delay Equations with Jumps

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Yan Li ◽  
Junhao Hu
Keyword(s):  
Brownian Motion ◽  
Numerical Analysis ◽  
Convergence Rate ◽  
Point Processes ◽  
Delay Equations ◽  
Infinite Dimensions ◽  
Partial Differential ◽  
Differential Delay ◽  
Differential Delay Equations ◽  
Exponential Integrator

We investigate the convergence rate of Euler-Maruyama method for a class of stochastic partial differential delay equations driven by both Brownian motion and Poisson point processes. We discretize in space by a Galerkin method and in time by using a stochastic exponential integrator. We generalize some results of Bao et al. (2011) and Jacob et al. (2009) in finite dimensions to a class of stochastic partial differential delay equations with jumps in infinite dimensions.

Stability in distribution of mild solutions to stochastic partial differential delay equations with jumps

2009 ◽  
Vol 465 (2107) ◽  
pp. 2111-2134 ◽  
Author(s):  
Jianhai Bao ◽  
Aubrey Truman ◽  
Chenggui Yuan
Keyword(s):  
Exponential Stability ◽  
Mild Solutions ◽  
Sufficient Conditions ◽  
Strong Solutions ◽  
Delay Equations ◽  
Infinite Dimensions ◽  
Partial Differential ◽  
Differential Delay ◽  
Differential Delay Equations ◽  
Stability In Distribution

The existence, uniqueness and some sufficient conditions for stability in distribution of mild solutions to stochastic partial differential delay equations with jumps are presented. The principle technique of our investigation is to construct a proper approximating strong solution system and carry out a limiting type of argument to pass on stability of strong solutions to mild ones. As a consequence, stability results of Basak et al . ( Basak et al . 1999 J. Math. Anal. Appl. 202 , 604–622) and Yuan et al . ( Yuan et al . 2003 Syst. Control Lett. 50 , 195–207) are generalized to cover a class of much more general stochastic partial differential delay equations with jumps in infinite dimensions. In contrast to the almost sure exponential stability in Ichikawa ( Ichikawa 1982 J. Math. Anal. Appl. 90 , 12–44) and Luo & Liu ( Luo & Liu 2008 Stoch. Proc. Appl. 118 , 864–895) and the moment exponential stability in Luo & Liu, we present a new result on the stability in distribution of mild solutions. Finally, an example is given to demonstrate the applicability of our work.

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