Mean-square exponential stability of stochastic theta methods for nonlinear stochastic delay integro-differential equations

This paper deals with the mean-square exponential stability of stochastic theta methods for nonlinear stochastic delay integro-differential equations. It is shown that the stochastic theta methods inherit the mean-square exponential stability property of the underlying system. Moreover, the backward Euler method is mean-square exponentially stable with less restrictions on the step size. In addition, numerical experiments are presented to confirm the theoretical results.

[1]  Toshiyuki Koto,et al.  Stability of θ-methods for delay integro-differential equations , 2003 .

[2]  Evelyn Buckwar.  The ⊝-Maruyama scheme for stochastic functional differential equations with distributed memory term * , 2004, Monte Carlo Methods Appl..

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

[4]  G. Milstein.  Numerical Integration of Stochastic Differential Equations , 1994 .

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

[6]  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..

[7]  Alexandra Rodkina,et al.  Almost sure asymptotic stability of drift-implicit θ-methods for bilinear ordinary stochastic differential equations in R 1 , 2005 .

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

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

[10]  Chengjian Zhang,et al.  Stability analysis of Runge-Kutta methods for nonlinear Volterra delay-integro-differential equations , 2004 .

[11]  V. B. Kolmanovskii,et al.  Stability of epidemic model with time delays influenced by stochastic perturbations 1 This paper was , 1998 .

[12]  Desmond J. Higham,et al.  Exponential mean square stability of numerical solutions to stochastic differential equations , 2003 .

[13]  X. Mao,et al.  Stochastic Differential Equations and Applications , 1998 .

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

[15]  Xuerong Mao,et al.  Almost sure exponential stability of numerical solutions for stochastic delay differential equations , 2010, Numerische Mathematik.

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

[17]  X. Mao,et al.  Exponential Stability of Stochastic Di erential Equations , 1994 .

[18]  Yi Shen,et al.  Stability of the split-step backward Euler scheme for stochastic delay integro-differential equations with Markovian switching , 2011 .

[19]  G. N. Milstein,et al.  Numerical Integration of Stochastic Differential Equations with Nonglobally Lipschitz Coefficients , 2005, SIAM J. Numer. Anal..

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

[21]  E. Hairer,et al.  Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems , 2010 .

[22]  L. Rogers,et al.  Complete Models with Stochastic Volatility , 1998 .

[23]  Mou-Hsiung Chang,et al.  The European option with hereditary price structures: Basic theory , 1999, Appl. Math. Comput..

[24]  P. Kloeden,et al.  Numerical Solution of Stochastic Differential Equations , 1992 .