Almost sure exponential stability of numerical solutions for stochastic delay differential equations with jumps

This paper deals with the almost sure exponential stability of the Euler-type methods for nonlinear stochastic delay differential equations with jumps by using the discrete semimartingale convergence theorem. It is shown that the explicit Euler method reproduces the almost sure exponential stability under an additional linear growth condition. By replacing the linear growth condition with the one-sided Lipschitz condition, the backward Euler method is able to reproduce the stability property.

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

[2]  Nicola Bruti-Liberati,et al.  Approximation of jump diffusions in finance and economics , 2007 .

[3]  Desmond J. Higham,et al.  Numerical methods for nonlinear stochastic differential equations with jumps , 2005, Numerische Mathematik.

[4]  Desmond J. Higham,et al.  Almost Sure and Moment Exponential Stability in the Numerical Simulation of Stochastic Differential Equations , 2007, SIAM J. Numer. Anal..

[5]  Desmond J. Higham,et al.  Asymptotic Stability of a Jump-Diffusion Equation and Its Numerical Approximation , 2008, SIAM J. Sci. Comput..

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

[7]  Chenggui Yuan,et al.  Numerical Solutions of Stochastic Differential Delay Equations with Jumps , 2009 .

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

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

[10]  R. Cont,et al.  Financial Modelling with Jump Processes , 2003 .

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

[12]  Changlin Mei,et al.  The semi-implicit Euler method for stochastic differential delay equation with jumps , 2007, Appl. Math. Comput..

[13]  Jiaowan Luo,et al.  Comparison principle and stability of Ito stochastic differential delay equations with Poisson jump and Markovian switching , 2006 .

[14]  X. Mao.  LaSalle-Type Theorems for Stochastic Differential Delay Equations , 1999 .

[15]  P. Kloeden,et al.  CONVERGENCE AND STABILITY OF IMPLICIT METHODS FOR JUMP-DIFFUSION SYSTEMS , 2005 .

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

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

[18]  X. Mao.  A note on the LaSalle-type theorems for stochastic differential delay equations , 2002 .

[19]  Li Ronghua,et al.  Convergence of numerical solutions to stochastic delay differential equations with jumps , 2006 .

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

[21]  Desmond J. Higham,et al.  On the Boundedness of Asymptotic Stability Regions for the Stochastic Theta Method , 2003 .

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