Variable step size control in the numerical solution of stochastic differential equations

SIAM Journal on Applied Mathematics

Volumn 57, Issue 5, 1997, Pages 1455-1484

  Gaines, J G ^a   Lyons, T J ^b  

^a UNIVERSITY OF EDINBURGH   (United Kingdom)

^b IMPERIAL COLLEGE LONDON   (United Kingdom)

Author keywords

L vy area; Numerical approximations; Stochastic differential equations; Variable step

Indexed keywords

APPROXIMATION THEORY; BROWNIAN MOVEMENT; CONVERGENCE OF NUMERICAL METHODS; ERROR ANALYSIS; THEOREM PROVING; TREES (MATHEMATICS);

LEVY AREA; STOCHASTIC DIFFERENTIAL EQUATIONS (SDE);

DIFFERENTIAL EQUATIONS;

EID: 0031258411     PISSN: 00361399     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036139995286515     Document Type: Article

References (23)

1. S. S. ARTEMEV AND I. O. SHKURKO, An algorithm of variable order and step based on methods of the Rozenbrok type, USSR Comp. Math, and Math. Phys., 26 (1986), pp. 193-195.
2. p theory of semimartingales, Ann. Probab., 9 (1981), pp. 49-89.
3. F. CASTELL AND J. G. GAINES, The ordinary differential equation approach to asymptotically efficient schemes for solution of stochastic differential equations, Ann. Inst. H. Poincaré Probab. Statist., 32 (1996), pp. 231-250.
4. J. M. C. CLARK, An efficient approximation for a class of stochastic differential equations, in Advances in Filtering and Optimal Stochastic Control, Proceedings of IFIP-WG7/1 Working Conference, Cocoyoc, Mexico, 1982, W. H. Fleming and L. G. Gorostiza, eds., Lecture Notes in Control and Information Sciences 42, Springer-Verlag, Berlin, 1982.
5. J. M. C. CLARK AND R. J. CAMERON, The maximum rate of convergence of discrete approximations for stochastic differential equations, in Stochastic Differential Systems, B. Grigelionis, ed., Lecture Notes in Control and Information Sciences 25, Springer-Verlag, Berlin, 1980.
6. O. FAURE, Numerical pathwise approximation of stochastic differential equations, Ecole Nationale des Ponts et Chaussées, Laboratoire de Mathématiques et Modélisation Report 9017, Noisy-le-Grand, France, 1990.
7. O. FAURE, Simulation du mouvement brownien et des diffusions, Ph.D. thesis, Ecole Nationale des Ponts et Chaussées, Noisy-le-Grand, France, 1991.
8. O. FAURE AND J. G. GAINES, Simulation trajectorielle des diffusions, in Probabilités Numériques, N. Bouleau and D. Talay, eds., INRIA, Rocquencourt, France, 1992, pp. 186-192.
9. J. G. GAINES, Numerical experiments with S(P)DE'S, in Stochastic Partial Differential Equations, A. M. Etheridge, ed., London Mathematical Society Lecture Note Series 216, Cambridge University Press, Cambridge, UK, 1995, pp. 55-71.
10. J. G. GAINES AND T. J. LYONS, Random generation of stochastic area integrals, SIAM J. Appl. Math., 54 (1994), pp. 1132-1146.
11. R. Z. HASMINSKII, Stochastic stability of differential equations, Sijthoff and Noordhoff, Groningen, the Netherlands, 1980.
12. I. KARATZAS AND S. E. SHREVE, Brownian Motion and Stochastic Calculus, Springer-Verlag, New York, 1988.
13. P. E. KLOEDEN AND E. PLATEN, Stratonovich and Itô stochastic Taylor expansions, Math. Nachr., 151 (1991), pp. 33-50.
14. P. E. KLOEDEN AND E. PLATEN, Numerical Solution of Stochastic Differential Equations, Applications of Mathematics 23, Springer-Verlag, New York, 1992.
15. P. LÉVY, Processus Stochastiques et Mouvement Brownien, Monographies des Probabilités, Gauthier-Villars, Paris, 1948.
16. T. LYONS, Differential Equations Driven by Rough Signals, preprint, 1996.
17. T. J. LYONS AND Z. M. QIAN, Flow of Diffeomorphisms Induced by a Geometric Multiplicative Functional, preprint, 1995.
18. N. J. NEWTON, An asymptotically efficient difference formula for solving stochastic differential equations, Stochastics, 19 (1986), pp. 175-206.
19. N. J. NEWTON, An efficient approximation for stochastic differential equations on the partition of symmetrical first passage times, Stochastics Stochastic Rep., 29 (1989), pp. 227-258.
20. N. J. NEWTON, Asymptotically efficient Runge-Kutta methods for a class of Itô and Stratonovich equations, SIAM J. Appl. Math., 51 (1991), pp. 542-567.
21. E. PLATEN AND W. WAGNER, On a Taylor formula for a class of Itô processes, Probab. Math. Statist., 3 (1982), pp. 37-51.
22. L. C. G. ROGERS AND D. WILLIAMS, Diffusions, Markov Processes and Martingales 2, Itô Calculus, John Wiley, New York, 1987.
23. D. TALAY, Simulation and numerical analysis of stochastic differential systems: A review, in Probabilistic Methods in Applied Physics, P. Krée and W. Wedig, eds., Lecture Notes in Physics 451, Springer-Verlag, New York, 1995, chap. 3, pp. 54-96.