Modified Euler scheme for the weak approximation of stochastic differential equations driven by the Wiener process
Authors:
S. V. Bodnarchuk and O. M. Kulyk
Translated by:
S. V. Kvasko
Journal:
Theor. Probability and Math. Statist. 99 (2019), 53-65
MSC (2010):
Primary 60H35; Secondary 60G51
DOI:
https://doi.org/10.1090/tpms/1079
Published electronically:
February 27, 2020
MathSciNet review:
3908655
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Additional Information
Abstract: A method for the weak approximation of solutions of stochastic differential equations driven by the Wiener process is considered in this paper.
References
- I. I. Gihman and A. V. Skorohod, Stokhasticheskie differentsial′nye uravneniya, Izdat. “Naukova Dumka”, Kiev, 1968 (Russian). MR 0263172
- Einar Hille and Ralph S. Phillips, Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, Vol. 31, American Mathematical Society, Providence, R. I., 1957. rev. ed. MR 0089373
- Peter E. Kloeden and Eckhard Platen, Numerical solution of stochastic differential equations, Applications of Mathematics (New York), vol. 23, Springer-Verlag, Berlin, 1992. MR 1214374
- Gisir\B{o} Maruyama, Continuous Markov processes and stochastic equations, Rend. Circ. Mat. Palermo (2) 4 (1955), 48–90. MR 71666, DOI https://doi.org/10.1007/BF02846028
- Remigijus Mikulevičius and Eckhard Platen, Time discrete Taylor approximations for Itô processes with jump component, Math. Nachr. 138 (1988), 93–104. MR 975202, DOI https://doi.org/10.1002/mana.19881380107
- G. N. Mil′šteĭn, A method with second order accuracy for the integration of stochastic differential equations, Teor. Verojatnost. i Primenen. 23 (1978), no. 2, 414–419 (Russian, with English summary). MR 0517998
- E. Platen, Zur zeitdiskreten Approximation von Itô prozessen, Diss. B., IMath, Akad. der Wiss. der DDR, Berlin (1984).
- Eckhard Platen and Wolfgang Wagner, On a Taylor formula for a class of Itô processes, Probab. Math. Statist. 3 (1982), no. 1, 37–51 (1983). MR 715753
- Denis Talay, Efficient numerical schemes for the approximation of expectations of functionals of the solution of a SDE and applications, Filtering and control of random processes (Paris, 1983) Lect. Notes Control Inf. Sci., vol. 61, Springer, Berlin, 1984, pp. 294–313. MR 874837, DOI https://doi.org/10.1007/BFb0006577
References
- I. I. Gihman and A. V. Skorohod, Stochastic Differential Equations, “Naukova Dumka”, Kiev, 1968; English transl. Springer, Berlin, 1972. MR 0263172
- E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups, American Mathematical Society, Providence, R. I., 1996. MR 0089373
- P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, Berlin, 1995. MR 1214374
- G. Maruyama, Continuous Markov processes and stochastic equations, Rendiconti del Circolo Matematico di Palermo 4 (1955), 48–90. MR 71666
- R. Mikulevicius and E. Platen, Time discrete Taylor approximations for Ito processes with jump component, Math. Nachr. 138 (1988), 93–104. MR 975202
- G. N. Milshtein, A method of second-order accuracy integration of stochastic differential equations, Teor. Veroyatnost. Primenen. 23 (1978), no. 2, 414–419; English transl. in Theory Probab. Appl. 23 (1978), no. 2, 396–401. MR 0517998
- E. Platen, Zur zeitdiskreten Approximation von Itô prozessen, Diss. B., IMath, Akad. der Wiss. der DDR, Berlin (1984).
- E. Platen and W. Wagner, On a Taylor formula for a class of Itô processes, Probab. Math. Statist. 3 (1982), no. 1, 37–51. MR 715753
- D. Talay, Efficient numerical schemes for the approximation of expectations of functionals of the solution of a SDE and applications, Lecture Notes in Control and Information Sciences 61 (1984), 294–313. MR 874837
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Additional Information
S. V. Bodnarchuk
Affiliation:
National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Peremogy Avenue, 37, Kyiv, 03056 Ukraine
Email:
sem_bodn@ukr.net
O. M. Kulyk
Affiliation:
Institute of Mathematics of National Academy of Science of Ukraine, Tereshchenkivska Street, 3, Kyiv, 01601 Ukraine
Email:
kulik@imath.kiev.ua
Keywords:
Stochastic differential equations,
Euler method,
weak approximation,
Hermite polynomials
Received by editor(s):
July 21, 2018
Published electronically:
February 27, 2020
Article copyright:
© Copyright 2020
American Mathematical Society
