×

zbMATH — the first resource for mathematics

Convergence and stability of numerical solutions to SDDEs with Markovian switching. (English) Zbl 1095.65005
The authors consider the numerical solution of stochastic delay differential equations (SDDEs) that are influenced by Markov switching processes. It clarifies the numerical stability and strong convergence of the Euler method.

MSC:
65C30 Numerical solutions to stochastic differential and integral equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Mao, X.; Matasov, A.; Piunovskiy, A.B., Stochastic differential delay equations with Markovian switching[J], Bernoulli, 6, 1, 73-90, (2000) · Zbl 0956.60060
[2] Mao, X., Robustness of stability of stochastic differential delay equations with Markovian switching[J], Sacta, 3, 1, 48-61, (2000)
[3] Shaikhet, L., Stability of stochastic hereditary systems with Markov switching[J], Stochastic processes, 2, 18, 180-184, (1996) · Zbl 0939.60049
[4] H. Gilsing, On the stability of the Euler scheme for an affine stochastic delay differential equation with time delay, Technical Report, Humboldt University, Berlin (in German) Diskussion Paper No. 20, SFB 373, 2001.
[5] Swishchuk, A.V.; Kazmerchuk, Y.I., Stability of stochastic differential delay ito’s equations with Poisson jumps and with Markovian switching. application to financial models, Teor. imovir. mat. stat., 63, 63, (2001)
[6] C.T.H. Baker, E. Buckwar, Introduction to the numerical analysis of stochastic delay differential equations, Technical Report, University of Manchester, Department of Mathematics, Numerical Analysis Report 345, 1999.
[7] Buckwar, E., Introduction to the numerical analysis of stochastic delay differential equations[J], J. comput. appl. math., 125, 297-307, (2000) · Zbl 0971.65004
[8] Kücher, U.; Platen, E., Strong discrete time approximation of stochastic differential equations with time delay[J], Math. comput. simul., 54, 189-205, (2000)
[9] Kücher, U.; Platen, E., Weak discrete time approximation of stochastic differential equations with time delay[J], Math. comput. simul., 59, 497-507, (2002) · Zbl 1001.65005
[10] Mao, X.; Sabanis, S., Numerical solutions of stochastic differential delay equtions under local Lipschitz condition[J], J. comput. appl. math., 151, 215-227, (2003) · Zbl 1015.65002
[11] Liu, M.; Cao, W.; Fan, Z., Convergence and stability of the semi-implicit Euler method for a linear stochastic differential delay equation[J], J. comput. appl. math., 170, 255-268, (2004) · Zbl 1059.65006
[12] Anderson, W.J., Continuous-time Markov chains[M], (1991), Springer Berlin
[13] Cao, W.; Liu, M.; Fan, Z., MS-stability of the Euler-Maruyama method for stochastic differential delay equations[J], J. appl. math. comput., 159, 127-135, (2004) · Zbl 1074.65007
[14] Yuan, C.; Mao, X., Convergence of the Euler-Maruyama method for stochastic differential equations with Markovian switching[J], Math. comput. simul., 64, 223-235, (2004) · Zbl 1044.65007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.