Convergence rates for adaptive weak approximation of stochastic differential equations

Stochastic Analysis and Applications

Volumn 23, Issue 3, 2005, Pages 511-558

  Moon, Kyoung Sook ^a   Szepessy, Anders ^b   Tempone, Raúl ^c,d   Zouraris, Georgios E ^e,f  

^a Bldg 142   (United States)

^b ROYAL INSTITUTE OF TECHNOLOGY   (Sweden)

^c UNIVERSITY OF TEXAS AT AUSTIN   (United States)

^d Instituto de Matematica y Estadistica Prof Ing Rafael Laguardia   (Uruguay)

^e UNIVERSITY OF CRETE   (Greece)

^f INSTITUTE OF COMPUTER SCIENCE   (Greece)

Author keywords

Adaptive mesh refinement algorithm; Almost sure convergence; Computational complexity; Monte Carlo method; Stochastic differential equations

Indexed keywords

EID: 20444378209     PISSN: 07362994     EISSN: 15329356     Source Type: Journal    
DOI: 10.1081/SAP-200056678     Document Type: Article

References (41)

1. Abdullaev, F. Kh., J. C. Bronski and G. Papanicolaou. 2000. Soliton perturbations and the random Kepler problem. Phys. D. 135:369-386.
2. Ainsworth M. and J. T. Oden. 1997. A posteriori error estimation in finite element analysis. Comput. Methods Appl. Mech. Engrg. 1-42.
3. Babuška, I. and W. C. Rheinboldt. 1978. Error estimates for adaptive finite element computations. SIAM J. Numer. Anal. 15:736-754.
4. Babuška, I. and M. Vogelius. 1984. Feedback and adaptive finite element solution of one-dimensional boundary value problems. Numer. Math. 44(1):75-102.
5. Becker, R., and R. Rannacher. 1996. A feed-back approach to error control in finite element methods: basic analysis and examples. East-West J. Numer. Math. (4)237-264.
6. Becker, R., and R. Rannacher. 2001. An optimal control approach to a posteriori error estimation in finite element methods. Acta Numerica 1-102.
7. Binev, P., W. Dahmen, and R. DeVore. 2004. Adaptive finite element methods with convergence rates. Numer. Math. 97(2):219-268.
8. Björk, T., A. Szepessy, R. Tempone, and G. Zouraris. 2002. Monte Carlo euler approximation if HJM term structure financial models, preprint.
9. Buchmann, F., and W. Petersen. 2002. Solving Dirichlet problems numerically using the Feynman-Kac representation. Research report no. 2002-01, ETH.
10. Cohen, A., W. Dahmen, and R. DeVore. 2000. Adaptive wavelet method for elliptic operator equations: convergence rates. Math. Comp 70(233):27-75.
11. Dahmen, W. 1997. Wavellet and multiscale methods for operator equations. Acta Numerica 6:55-228.
12. DeVore, R. A. 1998. Nonlinear approximation. Acta Numerica 51-150.
13. Durett, R. 1964. Probability: Theory and Examples. Belmont, CA: Duxbury Press.
14. Dzougotov, A. K.-S. Moon, E. von Schwerin, A. Szepessy, and R. Tempone. 2005. Adaptive Monte Carlo algorithms for stopped diffusion. Lecture Notes in Computational Science and Engineering. Heidelberg: Springer Verlag.
15. Eriksson, K., D. Estep, P. Hansbo, and C. Johnson. 1995. Introduction to adaptive methods for differential equations. Acta Numerica 105-158.
16. Fishman, G. S. 1996. Monte Carlo: Concepts, Algorithms, Applications. New York, NY: Springer-Verlag.
17. Glasserman, P. 2003. Monte Carlo Methods in Financial Engineering. New York, NY: Springer-Verlag.
18. Hammersley, J. M., and K. W. Morton. 1956. A new Monte Carlo technique: Antithetic variates. Proc. Cambridge Philos. Soc. 52:449-475.
19. Hofmann, N., P. Müller-Gronbach, and K. Ritter. 2000. Optimal approximation of stochastic differential equations by adaptive step-size control. Math. Comp. 69:1017-1034.
20. Hofmann, N., P. Müller-Gronbach, and K. Ritter. 2001. The optimal discretization of stochastic differential equations. J. Complexity 17:117-153.
21. Harrier, E., S. P. Norsett, and G. Wanner. 1993. Solving Ordinary Differential Equations. New York, NY: Springer-Verlag.
22. Johnson, C. and A. Szepessy. 1995. Adaptive finite element methods for conservation laws based on a posteriori error estimates. Comm. Pure Appl. Math. 48:199-234.
23. Jouini, C., J. Cvitanić, and M. Musiela. 2001. Option Pricing, Interest Rates and Risk Management. New York, NY: Cambridge University Press.
24. Karatzas, I., and S. E. Shreve. 1998. Brownian Motion and Stochastic Calculus, Graduate Texts in Mathematics 113. New York, NY: Springer-Verlag.
25. Kloeden, P. E., and E. Platen. 1992. Numerical solution of stochastic differential equations. Applications of Mathematics 23.
26. Majda, A., I. Timofeyev, and E. Vanden Eijnden. 2001. A mathematical framework for stochastic climate models. Comm. Pure Appl. Math. 54: 891-974.
27. Milstein, G. N. 1995. Numerical integration of stochastic differential equations, Mathematics and its Applications 313.
28. Moler. C. 1995. Random thoughts (http://www.mathworks.com/company/ newsletter/pdf/Cleve.pdf).
29. Moon, K.-S. 2001. Convergence Rates of Adaptive Algorithms for Deterministic and Stochastic Differential Equations, (Licentiate thesis, ISBN 91-7283-196-0, Royal Institute of Technology, 2001). (http://www.nada.kth.se/ ~moon/paper.html).
30. Moon, K.-S., A. Szepessy, R. Tempone, and G. E. Zouraris, 2001. Hyperbolic differential equations and adaptive numerics. In Theory and Numerics of Differential Equations, Durham 2000 eds. J. F. Blowey, J. P. Coleman and A. W. Craig. Universitext, Springer: Berlin, 231-280.
31. Moon, K.-S., A. Szepessy, R. Tempone, and G. E. Zouraris. 2003. A variational principle for adaptive approximation of ordinary differential equations. Numer. Math. 93:131-152.
32. Moon, K.-S., A. Szepessy, R. Tempone, and G. E. Zouraris. 2003. Convergence rates for adaptive approximation of ordinary differential equation. Numer. Math. 93:99-129.
33. Morin, R., H. Nochetto, and K. G. Sieber. 2000. Data oscillation and convergence of adaptive FEM. SIAM J. Numer. Anal. 38(2):466-488.
34. Müller-Gronbach, 2002. The optimal uniform approximation of systems of stochastic differential equations. Ann. Appl. Probab. 12(2):664-690.
35. Nualart, D. 1995. The Malliavin calculus and related topics. Probability and its Applications. New York: Springer-Verlag.
36. Øksendal, B. 1998. Stochastic Differential Equations, Fifth edition. Berlin: Springer-Verlag.
37. Press, W. H. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd edition. Cambridge, UK: Cambridge University Press.
38. Stevenson, R. 2003. An Optimal Adaptive Finite Element Method. Preprint no. 1271, Univ. of Utrecht.
39. Szepessy, A., R. Tempone, and G. E. Zouraris. 2001. Adaptive weak approximation of stochastic differential equations. Comm. Pure Appl. Math. 54(10):1169-1214
40. Talay, D. 1995. Simulation of stochastic differential systems. In Probabilistic Method in Applied Physics. (P. Krée, and W. Wedig, 54-96, Lecture notes in Physics, eds. 451. Berlin: Springer.
41. Talay, D. and L. Tubaro. 1990. Expansion of the global error for numerical schemes solving stochastic differential equation. Stochastic Anal. Appl. 8:483-509.