Optimal approximation of stochastic differential equations by adaptive step-size control

Mathematics of Computation

Volumn 69, Issue 231, 2000, Pages 1017-1034

  Hofmann, Norbert ^a   Müller Gronbach, Thomas ^b   Ritter, Klaus ^c  

^a UNIVERSITY OF ERLANGEN NUREMBERG   (Germany)

^b FREIE UNIVERSITÄT BERLIN   (Germany)

^c UNIVERSITY OF PASSAU   (Germany)

Author keywords

Adaption; Asymptotic optimality; Pathwise approximation; Step size control; Stochastic differential equations

Indexed keywords

EID: 0040364383     PISSN: 00255718     EISSN: None     Source Type: Journal    
DOI: 10.1090/s0025-5718-99-01177-1     Document Type: Article

References (35)

1. Bouleau, N. and Lépingle, D. (1994). Numerical Methods for Stochastic Processes. Wiley, New York. MR 95h:60090
2. Bouleau, N. and Lépingle, D. (1994). Numerical Methods for Stochastic Processes. Wiley, New York. MR 95h:60090
3. Cambanis, S. and Hu, Y. (1996). Exact convergence rate of the Euler-Maruyama scheme, with application to sampling design. Stochastics Stochastics Rep. 59, 211 240. MR 97k:60159
4. Cambanis, S. and Hu, Y. (1996). Exact convergence rate of the Euler-Maruyama scheme, with application to sampling design. Stochastics Stochastics Rep. 59, 211 240. MR 97k:60159
5. Clark, J. M. C. and Cameron, R. J. (1980). The maximum rate of convergence of discrete approximations. In Stochastic Differential Systems (B. Grigelionis, ed.) Lect. Notes Control Inf. Sci. 25. Springer, Berlin, 162-171. MR 82f:60133
6. Clark, J. M. C. and Cameron, R. J. (1980). The maximum rate of convergence of discrete approximations. In Stochastic Differential Systems (B. Grigelionis, ed.) Lect. Notes Control Inf. Sci. 25. Springer, Berlin, 162-171. MR 82f:60133
7. Faure, O. (1992). Simulation du mouvement brownien et des diffusions. Thèse ENPC, Paris.
8. Gaines, J. G. and Lyons, T. J. (1997). Variable step size control in the numerical solution of stochastic differential equations. SIAM J. Appl. Math. 57, 1455-1484.
9. MR 98m:60089
10. Kloeden, P. and Platen, E. (1992). Numerical Solution of Stochastic Differential Equations. Springer, Berlin. MR 94b:60069
11. Kloeden, P. and Platen, E. (1992). Numerical Solution of Stochastic Differential Equations. Springer, Berlin. MR 94b:60069
12. q-norm. J. Complexity 12, 47 57. MR 97b:41028
13. q-norm. J. Complexity 12, 47 57. MR 97b:41028
14. Mauthner, S. (1998). Step size control in the numerical solution of stochastic differential equations. Mathematische Preprintreihe der TU Darmstadt 1972; finally published in J. Comput. Appl. Math. 100 (1998), 93-109. CMP 99:05
15. Mauthner, S. (1998). Step size control in the numerical solution of stochastic differential equations. Mathematische Preprintreihe der TU Darmstadt 1972; finally published in J. Comput. Appl. Math. 100 (1998), 93-109. CMP 99:05
16. Mauthner, S. (1998). Step size control in the numerical solution of stochastic differential equations. Mathematische Preprintreihe der TU Darmstadt 1972; finally published in J. Comput. Appl. Math. 100 (1998), 93-109. CMP 99:05
17. Milstein, G. N. (1995). Numerical Integration of Stochastic Differential Equations. Kluwer, Dordrecht. MR 96e:65003
18. Milstein, G. N. (1995). Numerical Integration of Stochastic Differential Equations. Kluwer, Dordrecht. MR 96e:65003
19. Müller-Gronbach, T. (1996). Optimal design for approximating the path of a stochastic process. J. Statist. Planning Inf. 49, 371-385. MR 97h:62087
20. Müller-Gronbach, T. (1996). Optimal design for approximating the path of a stochastic process. J. Statist. Planning Inf. 49, 371-385. MR 97h:62087
21. Newton, N. J. (1990). An efficient approximation for stochastic differential equations on the partition of symmetrical first passage times. Stochastics Stochastics Rep. 29, 227-258. MR 91a:60152
22. Newton, N. J. (1990). An efficient approximation for stochastic differential equations on the partition of symmetrical first passage times. Stochastics Stochastics Rep. 29, 227-258. MR 91a:60152
23. Novak, E. and Ritter, K. (1993), Some complexity results for zero finding for univariate functions. J. Complexity 9, 15-40. MR 94c:65179
24. Novak, E. and Ritter, K. (1993), Some complexity results for zero finding for univariate functions. J. Complexity 9, 15-40. MR 94c:65179
25. Ritter, K. (1999). Average Case Analysis of Numerical Problems. Lect. Notes in Math., Springer, Berlin, to appear.
26. Sacks, J. and Ylvisaker, D. (1970). Statistical design and integral approximation. In Proc. 12th Bienn. Semin. Can. Math. Congr. (R. Pyke, ed.) Can. Math. Soc., Montreal, 115-136. MR 43:2806
27. Sacks, J. and Ylvisaker, D. (1970). Statistical design and integral approximation. In Proc. 12th Bienn. Semin. Can. Math. Congr. (R. Pyke, ed.) Can. Math. Soc., Montreal, 115-136. MR 43:2806
28. Shoji, I. (1998). Approximation of continuous time stochastic processes by a local linearization method. Math. Comp. 67, 287-298. MR 98e:65207
29. Shoji, I. (1998). Approximation of continuous time stochastic processes by a local linearization method. Math. Comp. 67, 287-298. MR 98e:65207
30. Su, Y. and Cambanis S. (1993). Sampling designs for estimation of random processes. Stochastic Processes Appl. 46, 47-89. MR 95d:62153
31. Su, Y. and Cambanis S. (1993). Sampling designs for estimation of random processes. Stochastic Processes Appl. 46, 47-89. MR 95d:62153
32. Talay, D. (1995). Simulation of stochastic differential systems. In Probabilistic Methods in Applied Physics (P. Krée, W. Wedig, eds.) Lecture Notes in Physics 451, Springer, Berlin., 54-96.
33. Traub, J. F., Wasilkowski, G. W. and Woźniakowski, H. (1988). Information-Based Complexity. Academic Press, New York. MR 90f:68085
34. Traub, J. F., Wasilkowski, G. W. and Woźniakowski, H. (1988). Information-Based Complexity. Academic Press, New York. MR 90f:68085
35. Wagner, W. and Platen, E. (1978). Approximation of Ito integral equations. Preprint ZIMM, Akad. Wiss. DDR, Berlin.