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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

$dx\left(t\right)=\mu \left(x\left(t\right),x\left(qt\right),r\left(t\right)\right)dt+\sigma \left(x\left(t\right),x\left(qt\right),r\left(t\right)\right)d{B}_{t},$

where $0 is a right-continuous Markov chain taking values in a finite set, ${B}_{t}$ is a standard $m$-dimensional Brownian motion.

##### MSC:
 65C30 Stochastic differential and integral equations 60H10 Stochastic ordinary differential equations 60H35 Computational methods for stochastic equations 34K50 Stochastic functional-differential equations 65L20 Stability and convergence of numerical methods for ODE 60J10 Markov chains (discrete-time Markov processes on discrete state spaces) 60J65 Brownian 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 $\theta$-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