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BIT Numerical Mathematics
Published in: BIT Numerical Mathematics 3/2014

01-09-2014

A weak second-order split-step method for numerical simulations of stochastic differential equations

Authors: C. Perret, W. P. Petersen

Published in: BIT Numerical Mathematics | Issue 3/2014

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Abstract

In an analogy from symmetric ordinary differential equation numerical integrators, we derive a three-stage, weak 2nd-order procedure for Monte-Carlo simulations of Itô stochastic differential equations. Our composite procedure splits each time step into three parts: an \(h/2\)-stage of trapezoidal rule, an \(h\)-stage martingale, followed by another \(h/2\)-stage of trapezoidal rule. In \(n\) time steps, an \(h/2\)-stage deterministic step follows another \(n-1\) times. Each of these adjacent pairs may be combined into a single \(h\)-stage, effectively producing a two-stage method with partial overlap between successive time steps.
previous article An energy-preserving exponentially-fitted continuous stage Runge–Kutta method for Hamiltonian systems
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Literature
1.
2.
3.
Coddington, E.A., Levinson, N.: Theory of Ordinary Differential Equations. McGraw-Hill, New York (1955)
4.
Gear, C.W.: Numerical Initial Value Problems in Ordinary Differential Equations. Prentice-Hall, Englewood Cliffs (1971)MATH
5.
Godunov, S.K., Ryabenkii, V.S.: Difference Schemes. Elsevier Publishers, Amsterdam (1987)MATH
6.
Hairer, E., Lubich, C., Wanner, E.: Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations. Springer, Berlin (2006)
7.
Hamming, R.W.: Numerical Methods for Scientists and Engineers. Dover Books, New York (1980)
8.
Higham, D.J., Mao, X., Stuart, A.M.: Strong convergence of Euler-type methods for nonlinear stochastic differential equations. SIAM J. Numer. Anal. 40(3), 1041–1063 (2002)
9.
Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes, pp. 218–220. Elsevier, North Holland (1981)
10.
Kloeden, P.E., Platen, E.: Numerical Solution of Stochastic Differential Equations. Springer, Berlin (1992)CrossRefMATH
11.
Milstein, G. N.: Numerical Solution of Stochastic Differential equations, (revised and translated version of 1988 Russian work published by Ural State University Press). Kluwer Academic (1995)
12.
Panneton, F., L’Ecuyer, P., Matsumoto, M.: Improved long-period generators based on linear recurrences modulo 2. ACM Trans. Math. Softw. (TOMS) 32(1), 1–16 (2006)MathSciNetCrossRef
13.
Petersen, W.P.: A general implicit splitting for stabilizing numerical simulations of Itô stochastic differential equations. SIAM J. Numer. Anal. 35(4), 1439–1451 (1998). (electronic)MathSciNetCrossRefMATH
14.
Stoer, J., Bulirsch, R.: Introduction to Numerical Analysis. Springer, New York (1980)CrossRef
15.
16.
Talay, D.: Discrétisation d’une équation différentielle stochastique et calcul approché d’espérances de fonctionnelles de la solution. RAIRO Modél. Math. Anal. Numér. 20(1), 141–179 (1986)MathSciNetMATH
17.
Metadata
Title
A weak second-order split-step method for numerical simulations of stochastic differential equations
Authors
C. Perret
W. P. Petersen
Publication date
01-09-2014
Publisher
Springer Netherlands
Published in
BIT Numerical Mathematics / Issue 3/2014
Print ISSN: 0006-3835
Electronic ISSN: 1572-9125
DOI
https://doi.org/10.1007/s10543-014-0482-4

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