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Approximations of Euler-Maruyama type for stochastic differential equations with Markovian switching, under non-Lipschitz conditions. (English) Zbl 1121.65011
The authors consider the stochastic differential equation with Markovian switching $dy(t)=f(y(t),r(t))dt+g(y(t),r(t))dw(t)$ where $$w(t)$$ is a $$d$$-dimensional Brownian motion, $$r(t)$$ is a right-continuous Markov chain, and $$w$$ and $$r$$ are independent. It is proved that the approximate solutions generated by the Euler-Maruyama numerical method converge in the $$L^1$$ sense and the $$L^2$$ sense under hypotheses sufficiently less restrictive to include some important equations arising in finance and engineering.

##### MSC:
 65C30 Numerical solutions to stochastic differential and integral equations 60H35 Computational methods for stochastic equations (aspects of stochastic analysis) 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 34F05 Ordinary differential equations and systems with randomness 60J65 Brownian motion
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##### References:
 [1] Anderson, W.J., Continuous-time Markov chains, (1991), Springer Berlin · Zbl 0721.60081 [2] Costa, O.L.V.; Boukas, K., Necessary and sufficient condition for robust stability and stabilizability of continuous-time linear systems with Markovian jumps, J. optim. theory appl., 99, 1155-1167, (1998) · Zbl 0919.93082 [3] Cox, J.C.; Ingersoll, J.E.; Ross, S.A., A theory of the term structure of interest rates, Econometrica, 53, 385-408, (1985) · Zbl 1274.91447 [4] Da Prato, G.; Zabczyk, J., Stochastic equations in infinite dimensions, (1992), Cambridge University Press Cambridge · Zbl 0761.60052 [5] Duarte, J., Evaluating an alternative risk preference in affine term structure models, Rev. financial studies, 17, 379-404, (2004) [6] Feng, X.; Loparo, K.A.; Ji, Y.; Chizeck, H.J., Stochastic stability properties of jump linear systems, IEEE trans. automat. control, 37, 38-53, (1992) · Zbl 0747.93079 [7] Heston, S., A closed-form solution for options with stochastic volatility with applications to bond and currency options, Rev. financial studies, 6, 327-343, (1993) · Zbl 1384.35131 [8] Higham, D.J.; Mao, X.; Stuart, A.M., Strong convergence of numerical methods for nonlinear stochastic differential equations, SIAM J. numer. anal., 40, 3, 1041-1063, (2002) · Zbl 1026.65003 [9] Kloeden, P.E.; Platen, E., Numerical solution of stochastic differential equations, (1992), Springer Berlin · Zbl 0925.65261 [10] Mao, X., Stochastic differential equations and applications, (1997), Horwood · Zbl 0874.60050 [11] Mao, X., Stability of stochastic differential equations with Markovian switching, Stochastic process. appl., 79, 45-67, (1999) · Zbl 0962.60043 [12] Mariton, M., Jump linear systems in automatic control, (1990), Marcel Dekker New York [13] Milstein, G.N., Numerical integration of stochastic differential equations, (1995), Kluwer Academic Publishers Dodrecht · Zbl 0810.65144 [14] Skorohod, A.V., Asymptotic methods in the theory of stochastic differential equations, (1989), American Mathematical Society Providence, RI [15] Yuan, C.; Mao, X., Convergence of the Euler-Maruyama method for stochastic differential equations with Markovian switching, Math. comput. simulation, 64, 223-235, (2004) · Zbl 1044.65007
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