Approximations of Numerical Method for Neutral Stochastic Functional Differential Equations with Markovian Switching

Stochastic systems with Markovian switching have been used in a variety of application areas, including biology, epidemiology, mechanics, economics, and finance. In this paper, we study the Euler-Maruyama (EM) method for neutral stochastic functional differential equations with Markovian switching. The main aim is to show that the numerical solutions will converge to the true solutions. Moreover, we obtain the convergence order of the approximate solutions.

[1]  Dhirendra Bahuguna,et al.  Approximations of solutions to neutral functional differential equations with nonlocal history conditions , 2006 .

[2]  Shaobo Zhou,et al.  Convergence of numerical solutions to neutral stochastic delay differential equations with Markovian switching , 2009 .

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

[4]  Yi Shen,et al.  Convergence of Numerical Approximation for Jump Models Involving Delay and Mean-Reverting Square Root Process , 2011 .

[5]  Yi Shen,et al.  The almost sure asymptotic stability and pth moment asymptotic stability of nonlinear stochastic delay differential systems with polynomial growth , 2012 .

[6]  Xuerong Mao,et al.  The improved LaSalle-type theorems for stochastic functional differential equations , 2006 .

[7]  X. Mao,et al.  Approximations of Euler-Maruyama type for stochastic differential equations with Markovian switching, under non-Lipschitz conditions , 2007 .

[8]  Ferenc Hartung,et al.  On existence, uniqueness and numerical approximation for neutral equations with state-dependent delays , 1997 .

[9]  Xuerong Mao,et al.  Convergence of the Euler-Maruyama method for stochastic differential equations with Markovian switching , 2004, Math. Comput. Simul..

[10]  M. Mariton,et al.  Jump Linear Systems in Automatic Control , 1992 .

[11]  Marino Zennaro,et al.  Stability analysis of one-step methods for neutral delay-differential equations , 1988 .

[12]  Xuerong Mao,et al.  Numerical Solutions of Neutral Stochastic Functional Differential Equations , 2008, SIAM J. Numer. Anal..

[13]  Lei Liu,et al.  Exponential Stability of Uncertain Stochastic Neural Networks with Markovian Switching , 2010, Neural Processing Letters.

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

[15]  Xuerong Mao,et al.  Numerical solutions of stochastic functional differential equations , 2003 .

[16]  Xuerong Mao,et al.  Stochastic Differential Equations With Markovian Switching , 2006 .

[17]  X. Mao.  Stochastic Functional Differential Equations with Markovian Switching , 2004 .

[18]  Kai Liu,et al.  On the exponential stability in mean square of neutral stochastic functional differential equations , 1999 .

[19]  Zdzislaw Jackiewicz,et al.  ONE-STEP METHODS OF ANY ORDER FOR NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS. , 1984 .

[20]  C. Yuan,et al.  Approximate solutions of stochastic differential delay equations with Markovian switching , 2006 .

[21]  Evelyn Buckwar,et al.  NUMERICAL ANALYSIS OF EXPLICIT ONE-STEP METHODS FOR STOCHASTIC DELAY DIFFERENTIAL EQUATIONS , 1975 .

[22]  Desmond J. Higham,et al.  Preserving exponential mean-square stability in the simulation of hybrid stochastic differential equations , 2007, Numerische Mathematik.

[23]  Yi Shen,et al.  Taylor approximation of the solutions of stochastic differential delay equations with Poisson jump , 2011 .

[24]  Yunkang Liu.  Numerical Solution of Implicit Neutral Functional Differential Equations , 1999 .

[25]  Josef Stoer,et al.  Numerische Mathematik 1 , 1989 .

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