Mean-square stability properties of an adaptive time-stepping SDE solver

Journal of Computational and Applied Mathematics

Volumn 194, Issue 2, 2006, Pages 245-254

  Lamba, H ^a   Seaman, T ^a  

^a GEORGE MASON UNIVERSITY   (United States)

Author keywords

Error control; Mean square stability; Milstein; Milstein type methods; Numerical integration; Stochastic differential equations; Variable step size

Indexed keywords

ADAPTIVE CONTROL SYSTEMS; DIFFERENTIAL EQUATIONS; DIFFUSION; MEASUREMENT ERRORS; NUMERICAL ANALYSIS; STOCHASTIC CONTROL SYSTEMS;

ERROR CONTROL; MEAN-SQUARE STABILITY; MILSTEIN; MILSTEIN-TYPE METHODS; NUMERICAL INTEGRATION; VARIABLE STEP-SIZE;

PROBLEM SOLVING;

EID: 33646496188     PISSN: 03770427     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cam.2005.07.007     Document Type: Article

References (33)

1. Artemiev S.S. Mean-square stability of numerical methods for solving stochastic differential equations. Russian Acad. Sci. Dokl. Math. 48 (1994) 539-542
2. Artemiev S.S. The mean-square stability of numerical methods for solving stochastic differential equations. Russian J. Numer. Anal. Math. Modeling 9 (1994) 405-416
3. Artemiev S.S., and Averina T.A. Numerical Analysis of Systems of Ordinary and Stochastic Differential Equations (1997), VSP, Utrecht
4. Bauer H. Probability Theory (1996), Walter de Gruyter, Berlin
5. Bryden A., and Higham D.J. On the boundedness of asymptotic stability regions for the stochastic theta method. BIT 43 1 (2003) 1-6
6. P.M. Burrage, Runge-Kutta methods for stochastic differential equations, Ph.D. Thesis, The University of Queensland, Brisbane, Queensland, Australia, January 1999.
7. Burrage K., and Burrage P. A variable stepsize implementation for stochastic differential equations. SIAM J. Sci. Comput. 24 (2003) 848-864
8. Burrage K., Burrage P.M., and Mitsui T. Numerical solutions of stochastic differential equations-implementation and stability issues. J. Comput. Appl. Math. 125 1-2 (2000) 171-182
9. Burrage K., and Tian T. A note on the stability properties of the Euler methods for solving stochastic differential equations. New Zealand J. Math. 29 (2000) 115-127
10. Gaines J.G., and Lyons T.J. Variable step size control in the numerical solution of stochastic differential equations. SIAM J. Appl. Math. 57 5 (1997) 1455-1484
11. D.F. Griffiths, The dynamics of some linear multistep methods with step-size control, in: Numerical Analysis 1987 (Dundee, 1987), Longman Sci. Tech., 1988, pp. 115-134.
12. Hall G. Equilibrium states of Runge-Kutta schemes. ACM Trans. Math. Software 11 (1985) 289-301
13. Hasḿinskii R.Z. Stochastic Stability of Differential Equations (1980), Sijthoff and Noordhoff, Alphen, Rijn
14. Hernandez D.B., and Spigler R. Convergence and stability of implicit Runge-Kutta methods for systems with multiplicative noise. BIT 33 (1993) 654-669
15. Higham D.J. A-stability and stochastic mean-square stability. BIT 40 2 (2000) 404-409
16. Higham D.J. Mean-square and asymptotic stability of the stochastic theta method. SIAM J. Numer. Anal. 38 3 (2000) 753-769
17. Hofmann N., Müller-Gronbach T., and Ritter K. Optimal approximation of stochastic differential equations by adaptive step-size control. Math. Comput. 69 231 (2000) 1017-1034
18. Kloeden P.E., and Platen E. Numerical Solution of Stochastic Differential Equations (1992), Springer, Berlin
19. Komori Y., Saito Y., and Mitsui T. Some issues in discrete approximate solution for stochastic differential equations. Comput. Math. Appl. 28 (1994) 269-278
20. Lamba H. An adaptive timestepping algorithm for stochastic differential equations. J. Comput. Appl. Math. 161 (2003) 417-430
21. H. Lamba, The effective stability of adaptive timestepping ODE solvers, SIAM J. Num. Anal., to appear.
22. H. Lamba, J. Mattingly, A.M. Stuart, An adaptive Euler-Maruyama scheme for SDEs: convergence and stability. Preprint.
23. Mannella R. Absorbing boundaries and optimal stopping in a stochastic differential equation. Phys. Lett. A 254 (1999) 257-262
24. Petersen W.P. Some experiments on numerical simulations of stochastic differential equations and a new algorithm. J. Comput. Phys. 113 1 (1994) 75-81
25. Petersen W.P. A general implicit splitting for stabilizing numerical simulations of Itô stochastic differential equations. SIAM J. Numer. Anal. 35 (1998) 1439-1451
26. Ryashko L.B., and Schurz H. Mean square stability analysis of some linear stochastic systems. Dynam. Systems Appl. 6 2 (1997) 165-189
27. Saito Y., and Mitsui T. T-stability of numerical schemes for stochastic differential equations. World Sci. Ser. Appl. Math. 2 (1993) 333-344
28. Saito Y., and Mitsui T. Stability analysis of numerical schemes for stochastic differential equations. SIAM J. Numer. Anal. 33 6 (1996) 2254-2267
29. Schurz H. Asymptotical mean square stability of an equilibrium point of some linear numerical solutions with multiplicative noise. Stochastic Anal. Appl. 14 (1996) 313-353
30. Schurz H. Stability, Stationarity, and Boundedness of Some Implicit Numerical Methods for Stochastic Differential Equations and Applications (1997), Logos Verlag Berlin, Berlin
31. Schurz H. The invariance of asymptotic laws of linear stochastic systems under discretization. Z. Angew. Math. Mech. 79 6 (1999) 375-382
32. H. Schurz, Numerical analysis of stochastic differential equations without tears, in: Handbook of Stochastic Analysis and Applications, Statistical Textbooks and Monography, vol. 163, Dekker, New York, 2002, pp. 237-359.
33. Stuart A.M., and Humphries A.R. The essential stability of local error control for dynamical systems. SIAM J. Numer. Anal. 32 (1995) 1940-1971