Stability of stochastic θ-methods for stochastic delay integro-differential equations

In this paper, we are concerned with the numerical stability of linear stochastic delay integro-differential equations (SDIDEs). A sufficient condition for mean square stability of the exact solution of a linear SDIDE with multiplicative noise is derived. Then the mean square stability of stochastic θ-methods is investigated, and it is shown that the numerical solution can reproduce the mean square stability of the exact solution under appropriate conditions. At last, we present some numerical experiments to support our conclusions.

[1]  X. Mao,et al.  Razumikhin-type theorems on exponential stability of stochastic functional differential equations , 1996 .

[2]  Siqing Gan,et al.  The split-step backward Euler method for linear stochastic delay differential equations , 2009 .

[3]  Mingzhu Liu,et al.  Convergence and stability of the semi-implicit Euler method for a linear stochastic differential delay equation , 2004 .

[4]  Krishnan Balachandran,et al.  Mean square stability of semi-implicit Euler method for linear stochastic differential equations with multiple delays and Markovian switching , 2008, Appl. Math. Comput..

[5]  Evelyn Buckwar,et al.  Exponential stability in p -th mean of solutions, and of convergent Euler-type solutions, of stochastic delay differential equations , 2005 .

[6]  R. Ash,et al.  Probability and measure theory , 1999 .

[7]  A. Bellen,et al.  Numerical methods for delay differential equations , 2003 .

[8]  Zhiyong Wang,et al.  An analysis of stability of milstein method for stochastic differential equations with delay , 2006, Comput. Math. Appl..

[9]  Xuerong Mao,et al.  Exponential stability of equidistant Euler-Maruyama approximations of stochastic differential delay equations , 2007 .

[10]  Thomas Müller-Gronbach,et al.  A modified Milstein scheme for approximation of stochastic delay differential equations with constant time lag , 2006 .

[11]  Wanrong Cao,et al.  MS-stability of the Euler-Maruyama method for stochastic differential delay equations , 2004, Appl. Math. Comput..

[12]  Xiaohua Ding,et al.  Convergence and stability of the semi-implicit Euler method for linear stochastic delay integro-differential equations , 2006, Int. J. Comput. Math..

[13]  Krishnan Balachandran,et al.  Mean-square stability of Milstein method for linear hybrid stochastic delay integro-differential equations , 2008 .

[14]  X. Mao,et al.  Numerical solutions of stochastic differential delay equations under local Lipschitz condition , 2003 .

[15]  Yoshihiro Saito,et al.  Stability Analysis of Numerical Schemes for Stochastic Differential Equations , 1996 .

[16]  La-sheng Wang,et al.  Convergence of numerical solutions to stochastic differential delay equations with Poisson jump and Markovian switching , 2007, Appl. Math. Comput..

[17]  Desmond J. Higham,et al.  Mean-Square and Asymptotic Stability of the Stochastic Theta Method , 2000, SIAM J. Numer. Anal..