Mean-square numerical methods for stochastic differential equations with small noises

SIAM Journal on Scientific Computing

Volumn 18, Issue 4, 1997, Pages 1067-1087

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

^a WEIERSTRASS INSTITUTE FOR APPLIED ANALYSIS AND STOCHASTICS   (Germany)

^b URAL FEDERAL UNIVERSITY   (Russian Federation)

Author keywords

Computer simulation; Small noises; Stochastic differential equations

Indexed keywords

LASER LANGEVIN EQUATION; LYAPUNOV EXPONENTS; MEAN SQUARE NUMERICAL METHODS; STOCHASTIC DIFFERENTIAL EQUATIONS;

COMPUTER SIMULATION; DIFFERENTIAL EQUATIONS; ERROR ANALYSIS; ESTIMATION; LYAPUNOV METHODS; RANDOM PROCESSES; THEOREM PROVING;

NUMERICAL METHODS;

EID: 0031173522     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/S1064827594278575     Document Type: Article

References (25)

1. C. W. GARDINER, Handbook of Stochastic Methods for Physics, Chemistry and Natural Sciences, Springer-Verlag, Berlin, 1983.
2. F. MOSS AND P. V. E. MCCLINTOCK, EDS., Noise in Nonlinear Dynamical Systems, Vol. I, II, III, Cambridge University Press, Cambridge, 1989.
3. G. N. MILSTEIN, Approximate integration of stochastic differential equations, Theory Probab. Appl., 19 (1974), pp. 557-562; A method of second-order accuracy integration of stochastic differential equations, Theory Probab. Appl., 23 (1978), pp. 396-401.
4. G. N. MILSTEIN, Approximate integration of stochastic differential equations, Theory Probab. Appl., 19 (1974), pp. 557-562; A method of second-order accuracy integration of stochastic differential equations, Theory Probab. Appl., 23 (1978), pp. 396-401.
5. W. WAGNER AND E. PLATEN, Approximation of Ito Integral Equations, Zentralinstitut für Mathematik und Mechanik, Akad. der Wiss. der DDR, Berlin, 1978, preprint.
6. D. TALAY, Efficient numerical schemes for the approximation of expectations of functionals of the solution of an SDE and applications, Lecture Notes in Control and Inform. Sci., 61 (1984), pp. 294-313; E. PARDOUX AND D. TALAY, Discretization and simulation of stochastic differential equations, Acta Appl. Math., 3 (1985), pp. 23-47.
7. D. TALAY, Efficient numerical schemes for the approximation of expectations of functionals of the solution of an SDE and applications, Lecture Notes in Control and Inform. Sci., 61 (1984), pp. 294-313; E. PARDOUX AND D. TALAY, Discretization and simulation of stochastic differential equations, Acta Appl. Math., 3 (1985), pp. 23-47.
8. G. N. MILSTEIN, Numerical Integration of Stochastic Differential Equations, Ural University Press, Sverdlovsk, 1988; Kluwer Academic Publishers, Norwell, MA, 1995.
9. P. E. KLOEDEN AND E. PLATEN, Numerical Solution of Stochastic Differential Equations, Springer-Verlag, Berlin, 1992.
10. G. N. MILSTEIN, A theorem on the order of convergence of mean-square approximations of solutions of systems of stochastic differential equations, Theory Probab. Appl., 32 (1987), pp. 738-741.
11. G. N. MILSTEIN AND M. V. TRET'YAKOV, Numerical solution of differential equations with colored noise, J. Statist. Phys., 77 (1994), pp. 691-715.
12. R. MANNELA AND V. PALLESCHI, Fast and precise algorithm for computer simulation of stochastic differential equations, Phys. Rev. A, 40 (1989), pp. 3381-3386; R. F. FOX, Second-order algorithm for the numerical integration of colored-noise problems, Phys. Rev. A, 43 (1991), pp. 2649-2654.
13. R. MANNELA AND V. PALLESCHI, Fast and precise algorithm for computer simulation of stochastic differential equations, Phys. Rev. A, 40 (1989), pp. 3381-3386; R. F. FOX, Second-order algorithm for the numerical integration of colored-noise problems, Phys. Rev. A, 43 (1991), pp. 2649-2654.
14. R. Z. HASMINSKI, Stochastic Stability of Differential Equations, Sijthoff & Noordhoff, Groningen, the Netherlands, 1980.
15. L. ARNOLD AND V. WIHSTUTZ, EDS., Lyapunov Exponents, Lecture Notes in Math., V. 1186, Springer-Verlag, Berlin, 1986; E. PARDOUX AND V. WIHSTUTZ, Lyapunov exponent and rotation number of two-dimensional linear stochastic systems with small diffusion, SIAM J. Appl. Math., 48 (1988), pp. 442-457.
16. L. ARNOLD AND V. WIHSTUTZ, EDS., Lyapunov Exponents, Lecture Notes in Math., V. 1186, Springer-Verlag, Berlin, 1986; E. PARDOUX AND V. WIHSTUTZ, Lyapunov exponent and rotation number of two-dimensional linear stochastic systems with small diffusion, SIAM J. Appl. Math., 48 (1988), pp. 442-457.
17. D. TALAY, Approximation of upper Lyapunov exponents of bilinear stochastic differential equations, SIAM J. Numer. Anal., 28 (1991), pp. 1141-1164.
18. 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.
19. W. H. PRESS, W. T. VETTERLING, S. A. TEUKOLSKY, AND B. P. FLANNERY, Numerical Recipes, Cambridge University Press, Cambridge, 1986.
20. R. GRAHAM, Hopf bifurcation with fluctuating control parameter, Phys. Rev. A, 25 (1982), pp. 3234-3258; A. SCHENZLE AND H. BRAND, Multiplicative stochastic processes in statistical physics, Phys. Rev. A, 20 (1979), pp. 1628-1647.
21. R. GRAHAM, Hopf bifurcation with fluctuating control parameter, Phys. Rev. A, 25 (1982), pp. 3234-3258; A. SCHENZLE AND H. BRAND, Multiplicative stochastic processes in statistical physics, Phys. Rev. A, 20 (1979), pp. 1628-1647.
22. H. RISKEN, The Fokker-Planck Equation, Springer-Verlag, Berlin, 1984.
23. S. N. DIXIT AND P. S. SAHNI, Nonlinear stochastic processes driven by colored noise: Application to dye-laser statistics, Phys. Rev. Lett., 50 (1983), pp. 1273-1276; R. F. FOX AND R. ROY, Steady-state analysis of strongly colored multiplicative noise in a dye laser, Phys. Rev. A, 35 (1987), pp. 1838-1842.
24. S. N. DIXIT AND P. S. SAHNI, Nonlinear stochastic processes driven by colored noise: Application to dye-laser statistics, Phys. Rev. Lett., 50 (1983), pp. 1273-1276; R. F. FOX AND R. ROY, Steady-state analysis of strongly colored multiplicative noise in a dye laser, Phys. Rev. A, 35 (1987), pp. 1838-1842.
25. F. MOSS, A. BULSARA, AND M. SHLESINGER, EDS., Proceedings of the NATO advanced research workshop: Stochastic resonance in physics and biology, J. Statist. Phys., 70 (1993), Nos. 1/2.