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Higher-Order Weak Approximation of Ito Diffusions by Markov Chains

Published online by Cambridge University Press:  27 July 2009

Eckhaard Platen
Eckhaard Platen
Affiliation:
Karl-Weierstrass-Institute of Mathematics Mohrenstr. 39 1086 Berlin, Germany
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Abstract

This paper proposes a method that allows the construction of discrete-state Markov chains approximating an Ito-diffusion process. The transition probabilities of the Markov chains are chosen in such a way that functionals converge with a desired weak order with respect to vanishing step size under sufficient smoothness assumptions.

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Type
Articles
Information
Probability in the Engineering and Informational Sciences , Volume 6 , Issue 3 , July 1992 , pp. 391 - 408
Copyright
Copyright © Cambridge University Press 1992

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References

Billingsley, P. (1968). Convergence of probability measures. New York: Wiley.Google Scholar
DiMasi, G. & Rungaldier, W. (1981). An approximation to optimal nonlinear filtering with discontinuous observations. Stochastic systems: The mathematics of filtering and identification and applications. Proceedings of the NATO Advanced Study Institute 583–590.Google Scholar
Ikeda, N. & Watanabe, S. (1981). Stochastic differential equations and diffusion processes. Amsterdam: Elsevier (North-Holland) (1989, 2 ed).Google Scholar
Jacod, J. & Shirjajev, A.N. (1987). Limit theorems for stochastic processes. New York: Springer- Verlag.CrossRefGoogle Scholar
Kloeden, P.E. & Platen, E. (1991). The numerical solution of stochastic differential equations. In Applications of mathematics. New York: Springer-Verlag.Google Scholar
Krylov, N.y. (1980). Controlled diffusion processes. New York: Springer-Verlag.CrossRefGoogle Scholar
Kushner, H.J. (1977). Probability met hods for approximations in stochastic control and for elliptic equations. New York: Academic Press.Google Scholar
Kushner, H. J. & DiMasi, G. (1978). Approximation for functionals and optimal control on jump diffusion processes. Journal of Mathematical Analysis Applications 63: 772–800.CrossRefGoogle Scholar
Mikulevicius, R. (1983). On some properties of solutions of stochastic differential equations. Lietuvos Matematikos Rinkinys 4: 18–31.Google Scholar
Mikulevicius, R. & Platen, E. (1988). Time discrete Taylor approximations for Ito processes with jump component. Math. Nachr. 138: 93–104.CrossRefGoogle Scholar
Milstein, G.N. (1978). A method of second-order accuracy for the integration of stochastic differential equations. Theory of Probability and Applications 23: 396–401.CrossRefGoogle Scholar
Platen, E. (1982). A generalized Taylor formula for solutions of stochastic differential equations. Sankhya 44A: 163–172.Google Scholar
Platen, E. (1984). Zur zeitdiskreten Approximation von Itoprozessen. Diss. B., lMath, Akad. der. Wiss. d. DDR, Berlin.Google Scholar
Platen, E. & Rebolledo, R. (1985). Weak convergence of semimartingales and discretization methods. Stochastic Processes and Their Applications 20: 41–58.CrossRefGoogle Scholar
Platen, E. & Wagner, W. (1982). On a Taylor formula for a class of Ito processes. Pro bability of Mathematical Statistics 3: 37–51.Google Scholar
Talay, D. (1984). Efficient numerical schemes for the approximation of expectation of functionals of the solution of an SDE and application. Springer Lecture Notes in Control and Information Sciences 61: 294–313.CrossRefGoogle Scholar