Delay dependent asymptotic mean square stability analysis of the stochastic exponential Euler method

Abstract This paper is concerned with the delay dependent stability of the stochastic exponential Euler method for stochastic delay differential equations and stochastic delay partial differential equations. By using root locus technique, the necessary and sufficient condition of the numerical delay dependent stability of the method is derived for a class of stochastic delay differential equations and it is shown that the stochastic exponential Euler method can fully preserve the asymptotic mean square stability of the underlying system. Furthermore, we investigate the delay dependent stability of the semidiscrete and fully discrete systems for a linear stochastic delay partial differential equation. The necessary and sufficient condition for the delay dependent stability of the semidiscrete system based on the standard central difference scheme in space is given. Based on this condition, the delay dependent stability of the fully discrete system by using the stochastic exponential Euler method in time is studied. It is shown that the fully discrete scheme can inherit the delay dependent stability of the semidiscrete system completely. At last, some numerical experiments are given to validate our theoretical results.

[1]  Chiping Zhang,et al.  The Convergence and MS Stability of Exponential Euler Method for Semilinear Stochastic Differential Equations , 2012 .

[2]  Wanrong Cao,et al.  On exponential mean-square stability of two-step Maruyama methods for stochastic delay differential equations , 2013, J. Comput. Appl. Math..

[3]  Wanrong Cao,et al.  Split-step θ-method for stochastic delay differential equations , 2014 .

[4]  Qian Guo,et al.  Almost sure exponential stability of an explicit stochastic orthogonal Runge-Kutta-Chebyshev method for stochastic delay differential equations , 2015 .

[5]  Xiaojie Wang,et al.  An Exponential Integrator Scheme for Time Discretization of Nonlinear Stochastic Wave Equation , 2013, J. Sci. Comput..

[6]  Qian Guo,et al.  Asymptotic mean-square stability of explicit Runge-Kutta Maruyama methods for stochastic delay differential equations , 2016, J. Comput. Appl. Math..

[7]  Peng Hu,et al.  Delay dependent stability of stochastic split-step θ methods for stochastic delay differential equations , 2018, Appl. Math. Comput..

[8]  A. Ostermann,et al.  A Class of Explicit Exponential General Linear Methods , 2006 .

[9]  Chengjian Zhang,et al.  A derivative-free explicit method with order 1.0 for solving stochastic delay differential equations , 2013, J. Comput. Appl. Math..

[10]  Chengming Huang,et al.  Delay-dependent stability of high order Runge–Kutta methods , 2009, Numerische Mathematik.

[11]  Salah-Eldin A. Mohammed,et al.  Discrete-time approximations of stochastic delay equations: The Milstein scheme , 2004 .

[12]  Siqing Gan,et al.  θ-Maruyama methods for nonlinear stochastic differential delay equations , 2015 .

[13]  Chengming Huang.  Mean square stability and dissipativity of two classes of theta methods for systems of stochastic delay differential equations , 2014, J. Comput. Appl. Math..

[14]  Le Yi Wang,et al.  Stochastic Consentability of Linear Systems With Time Delays and Multiplicative Noises , 2018, IEEE Transactions on Automatic Control.

[15]  A. Longtin,et al.  Small delay approximation of stochastic delay differential equations , 1999 .

[16]  Aiguo Xiao,et al.  Stability of analytical and numerical solutions of nonlinear stochastic delay differential equations , 2014, J. Comput. Appl. Math..

[17]  M. Riedle,et al.  Geometric Brownian motion with delay: mean square characterisation , 2007, math/0703837.

[18]  Siqing Gan,et al.  Delay-dependent stability analysis of numerical methods for stochastic delay differential equations , 2012, J. Comput. Appl. Math..

[19]  Nicola Guglielmi,et al.  Delay dependent stability regions of Θ-methods for delay differential equations , 1998 .

[20]  J. T. Ngnotchouye,et al.  Weak convergence for a stochastic exponential integrator and finite element discretization of stochastic partial differential equation with multiplicative & additive noise , 2016 .

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

[22]  M. Carletti.  Numerical solution of stochastic differential problems in the biosciences , 2006 .

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

[24]  Ling Zhang.  Convergence and stability of the exponential Euler method for semi-linear stochastic delay differential equations , 2017, Journal of inequalities and applications.

[25]  Mari Paz Calvo,et al.  A class of explicit multistep exponential integrators for semilinear problems , 2006, Numerische Mathematik.

[26]  Yu Zhang,et al.  Stability of Exponential Euler Method for Stochastic Systems under Poisson White Noise Excitations , 2014 .

[27]  Christopher T. H. Baker,et al.  Computing stability regions—Runge-Kutta methods for delay differential equations , 1994 .

[28]  E. Samiei,et al.  On Lyapunov stability of scalar stochastic time-delayed systems , 2013 .

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

[30]  M. Hochbruck,et al.  Exponential Runge--Kutta methods for parabolic problems , 2005 .

[31]  T. Shardlow,et al.  The Milstein Scheme for Stochastic Delay Differential Equations Without Using Anticipative Calculus , 2012 .

[32]  Marlis Hochbruck,et al.  Exponential Rosenbrock-Type Methods , 2008, SIAM J. Numer. Anal..

[33]  Evelyn Buckwar,et al.  Multi-Step Maruyama Methods for Stochastic Delay Differential Equations , 2007 .

[34]  Leonid Shaikhet.  Lyapunov Functionals and Stability of Stochastic Functional Differential Equations , 2013 .

[35]  G. Lord,et al.  Stochastic exponential integrators for finite element discretization of SPDEs for multiplicative and additive noise , 2011, 1103.1986.

[36]  Stefan Vandewalle,et al.  An Analysis of Delay-Dependent Stability for Ordinary and Partial Differential Equations with Fixed and Distributed Delays , 2004, SIAM J. Sci. Comput..

[37]  K. Burrage,et al.  A stochastic exponential Euler scheme for simulation of stiff biochemical reaction systems , 2014, BIT Numerical Mathematics.

[38]  Peter E. Kloeden,et al.  The exponential integrator scheme for stochastic partial differential equations: Pathwise error bounds , 2011, J. Comput. Appl. Math..

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

[40]  A note on the analysis of asymptotic mean-square stability properties for systems of linear stochastic delay differential equations , 2013 .

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

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