Numerical methods in the weak sense for stochastic differential equations with small noise

SIAM Journal on Numerical Analysis

Volumn 34, Issue 6, 1997, Pages 2142-2167

  Milstein, G N ^a   Tret'yakov, M V ^b  

^a Weierstra Inst Angew Anal/S   (Germany)

^b URAL FEDERAL UNIVERSITY   (Russian Federation)

Author keywords

Computer simulation; Monte carlo methods; Small noise; Weak approximation

Indexed keywords

EID: 0001625720     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036142996278967     Document Type: Article

References (17)

1. L. ARNOLD AND V. WIHSTUTZ, EDS., Lyapunov Exponents, Lecture Notes in Mathematics 1186, Springer-Verlag, Berlin, 1986, pp. 1-26.
2. E. I. AUSLENDER AND G. N. MILSTEIN, Asymptotic expansions of the Lyapunov index for linear stochastic systems with small noises, Prikl. Mat. Mekh., 46 (1982), pp. 358-365.
3. M. I. FREIDLIN AND A. D. WENTZELL, Random Perturbations of Dynamical Systems, Springer-Verlag, Berlin, 1984.
4. R. Z. HASMINSKI, Stochastic Stability of Differential Equations, Sijthoff & Noordhoff, Gröninger, the Netherlands, 1980.
5. P. E. KLOEDEN AND E. PLATEN, Numerical Solution of Stochastic Differential Equations, Springer-Verlag, Berlin, 1992.
6. G. N. MILSTEIN, A method of second-order accuracy integration of stochastic differential equations, Theory Probab. Appl., 23 (1978), pp. 396-401.
7. G. N. MILSTEIN, Weak approximation of solutions of systems of stochastic differential equations, Theory Probab. Appl., 30 (1985), pp. 750-766.
8. G. N. MILSTEIN, Numerical Integration of Stochastic Differential Equations, Ural University Press, Sverdlovsk, Russia, 1988; Kluwer Academic Publishers, Norwell, MA, 1995.
9. G. N, MILSTEIN AND M. V. TRET'YAKOV, Mean-square numerical methods for stochastic differential equations with small noises, SIAM J. Sci. Comput., 18 (1997), pp. 1067-1087.
10. G. N. MILSTEIN AND M. V. TRET'YAKOV, Numerical solution of differential equations with colored noise, J. Stat. Phys., 77 (1994), pp. 691-715.
11. G. N. MILSTEIN AND M. V. TRET'YAKOV, Weak approximation for stochastic differential equations with small noises, Preprint No. 123, Weierstraß-Institut für Angewandte Analysis und Stochastik, Berlin, 1994.
12. E. PARDOUX AND D. TALAY, Discretization and simulation of stochastic differential equations, Acta Appl. Math., 3 (1985), pp. 23-47.
13. W. H. PRESS, S. A. TEUKOLSKY, W. T. VETTERLING, AND B. P. FLANNERY, Numerical Recipes in Fortran: The Art of Scientific Computing, Cambridge University Press, Cambridge, UK, 1992.
14. D. TALAY, Efficient numerical schemes for the approximation of expectations of functional of the solution of an SDE and applications, Lecture Notes in Control and Inform. Sci. 61, Springer-Verlag, Berlin, 1984, pp. 294-313.
15. D. TALAY, Approximation of upper Lyapunov exponents of bilinear stochastic differential equations, SIAM J. Numer. Anal., 28 (1991), pp. 1141-1164.
16. D. TALAY AND L. TUBARO, Expansion of the global error for numerical schemes solving stochastic differential equations, Stochastic Anal. Appl., 8 (1990), pp. 483-509.
17. W. WAGNER AND E. PLATEN, Approximation of Ito Integral Equations, preprint, Zentralinstitut für Mathematik und Mechanik, Akad. der Wiss. der DDR, Berlin, 1978.