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Convergence of numerical solutions to stochastic pantograph equations with Markovian switching. (English) Zbl 1206.65023

The authors investigate the convergence of the Euler method for n-dimensional stochastic pantograph equations with Markovian switching of the form

dx(t)=μ(x(t),x(qt),r(t))dt+σ(x(t),x(qt),r(t))dB t ,

where 0<q<1,r(t) is a right-continuous Markov chain taking values in a finite set, B t is a standard m-dimensional Brownian motion.

MSC:
65C30Stochastic differential and integral equations
60H10Stochastic ordinary differential equations
60H35Computational methods for stochastic equations
34K50Stochastic functional-differential equations
65L20Stability and convergence of numerical methods for ODE
60J10Markov chains (discrete-time Markov processes on discrete state spaces)
60J65Brownian motion
References:
[1]J.A.D. Appleby, E. Buckwar, Sufficient Condition for Polynomial Asymptotic Behavior of the Stochastic Pantograph Equation. lt;http://www.dcu.ie/maths/research/preprint.shtmlgt;.
[2]Baker, C. T. H.; Buckwar, E.: Continuous θ-methods for the stochastic pantograph equation, Electron. trans. Numer. anal. 11, 131-151 (2000) · Zbl 0968.65004 · doi:emis:journals/ETNA/vol.11.2000/pp131-151.dir/pp131-151.html
[3]Fan, Zhencheng; Liu, Mingzhu; Cao, Wanrong: Existence and uniqueness of the solutions and convergence of semi-implicit Euler methods for stochastic pantograph equations, J. math. Anal. appl. 325, 1142-1159 (2007) · Zbl 1107.60030 · doi:10.1016/j.jmaa.2006.02.063
[4]Mao, Xuerong; Matasov, Alexander; Piunovskiy, Aleksey B.: Stochastic differential delay equations with Markovian switching, Bernoulli 6, No. 1, 73-90 (2000) · Zbl 0956.60060 · doi:10.2307/3318634
[5]Mao, Xuerong: Robustness of stability of stochastic differential delay equations with Markovian switching, Sacta 3, No. 1, 48-61 (2000)
[6]Mao, X.: Stochastic differential equations and applications, (1997)
[7]Ronghua, Li; Hongbing, Meng; Qin, Chang: Exponential stability of numerical solutions to sddes with Markovian switching, Appl. math. Comput. 174, 1302-1313 (2006) · Zbl 1105.65010 · doi:10.1016/j.amc.2005.05.037
[8]Ronghua, Li; Yingmin, Hou: Convergence and stability of numerical solutions to sddes with Markovian switching, Appl. math. Comput. 175, 1080-1091 (2006) · Zbl 1095.65005 · doi:10.1016/j.amc.2005.08.026
[9]Shaikhet, L.: Stability of stochastic hereditary systems with Markov switching, Stochast. process. 2, No. 18, 180-184 (1996) · Zbl 0939.60049
[10]Yuan, Chenggui; Mao, Xuerong: Convergence of the Euler – Maruyama method for stochastic differential equations with Markovian switching, Math. comput. Simulat. 64, 223-235 (2004) · Zbl 1044.65007 · doi:10.1016/j.matcom.2003.09.001