Weak approximations. A Malliavin calculus approach

Mathematics of Computation

Volumn 70, Issue 233, 2001, Pages 135-172

  Kohatsu Higa, Arturo ^a  

^a UNIVERSITAT POMPEU FABRA   (Spain)

Author keywords

Boundary conditions; Numerical analysis; Stochastic differential equations; Weak approximation

Indexed keywords

EID: 0035625675     PISSN: 00255718     EISSN: None     Source Type: Journal    
DOI: 10.1090/s0025-5718-00-01201-1     Document Type: Article

References (34)

1. M.B. Adams, A.S. Willsky and B.C. Levy, Linear estimation of boundary value stochastic processes, IEEE Trans. Automat. Control AC-29 (1984) 803-821. MR 85k:93064; MR 86a:93135
2. H. Ahn and A. Kohatsu-Higa, The Euler scheme for anticipating stochastic differential equations, Stochastics and Stochastics Rep. 54 (1995) 247-269. MR 97b:60098
3. V. Bally and D. Talay, The law of the Euler scheme for stochastic differential equations I: convergence rate of the distribution function, Probab. Theory Related Fields 104 (1996) 43-60. MR 96k:60136
4. V. Bally and D. Talay, The law of the Euler scheme for stochastic differential equations: II. Convergence rate of the density, Monte Carlo Methods Appl. 2 (1996) 93-128. MR 97k:60157
5. M. Bossy and D. Talay, A Stochastic Particle Method for the McKean-Vlasov and Burgers equations, Math. Comp. 66 (1997) 157-192. MR 97c:60233
6. M. Bossy and D. Talay, Convergence rate for the approximation of the limit law of weakly interacting particles: application to the Burgers equation, Ann. Appl. Probab. 6 (1996) 818-861. MR 97k:60158
7. R. Bellman and G.M. Wing, An introduction to invariant imbedding, Wiley, New York, 1976. MR 52:12558
8. C. Donati-Martin, Equations différentielles stochastiques dans ℝ avec conditions aux bords, Stochastics Stochastics Rep. 35 (1991) 143-173. MR 92h:60091
9. N. Dunford and L. Schwartz, Linear Operators, Part II, Pure and Applied Mathematics, A Series of Texts and Monographs, Intescience Pub., 1964. MR 32:6181
10. M. Ferrante, A. Kohatsu-Higa and M. Sanz-Solé, Strong approximations for stochastic differential equations with boundary conditions, Stochastic Processes Appl. 61 (1996) 323-337. MR 97b:60099
11. J.P. Fouque and E. Merzbach, A limit theorem for linear boundary value problems in random media, Ann. Appl. Probab. 4 (1994) 549-569. MR 95e:60063
12. L. E. Fraenkel, Formulae for high derivatives of composite functions, Math. Proc. Cambridge Philos. Soc. 83 (1978) 159-165. MR 58:6124
13. J. Garnier, Stochastic invariant imbedding. Application to stochastic differential equations with boundary conditions, Probab. Theory Related Fields 102 (1995) 249-271. MR 96k:60138
14. Y. Hu and S. Watanabe, Donsker delta functions and approximations of heat kernels by the time discretization method, J. Math. Kyoto Univ. 36 (1996) 499-518. MR 97k:60154
15. H. Keller, Numerical Methods for Two-Point Boundary-Valued Problems, Blaisdell, 1968. MR 37:6038
16. P.E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Applications of Mathematics 23, Springer, New York, 1992. MR 94b:60069
17. A. Kohatsu-Higa, High Itô-Taylor approximations to heat kernels, J. Math. Kyoto Univ. 37 (1997) 129-150 (1997). MR 98c:60060
18. A. Kohatsu-Higa and S. Ogawa, Weak rate of convergence for an Euler scheme of nonlinear SDE's, Monte Carlo Methods Appl. 3 (1997) 327-345. MR 98i:60053
19. A. Kohatsu-Higa and M. Sanz-Solé, Existence and regularity of the density for solutions to stochastic differential equations with boundary conditions, Stochastics Stochastics Rep. 60 (1997) 1-22. MR 98d:60111
20. A. Krener, Reciprocal processes and the stochastic realization problem for acausal system, Modelling, identification and robust control, C.I. Brynes and E. Liudquist, eds., Elsevier/North Holland, Amsterdam, New York, 1986, pp. 197-209. MR 89c:93065.
21. H. Kwakernaak, Periodic linear differential stochastic processes, SIAM J. Control Optim. 13 (1975) 400-413. MR 51:15114
22. G.N. Milshtein, A method of second order accuracy integration of stochastic differential equations Theor. Probab. Appl. 23 (1978) 396-401. MR 58:24549
23. D. Nualart, The Malliavin Calculus and Related Topics, Springer Verlag, 1995. MR 96k:60130
24. D. Nualart, Analysis on Wiener space and anticipating stochastic calculus, Lecture Notes in Mathematics vol. 1690, Ecole d'Eté de Probabilités de Saint-Flour XXV, Springer, Berlin, 1998, 123-227. MR 99k:60144
25. D. Nualart and E. Pardoux. Boundary value problems for stochastic differential equations, Ann. Probab. 19 (1991) 1118-1144. MR 92j:60072
26. D. Nualart and E. Pardoux, Second order stochastic differential equations with Dirichlet boundary conditions, Stochastic Processes Appl. 39 (1991) 1-24. MR 93b:60123
27. D. Ocone and E. Pardoux, Linear Stochastic Differential Equations with boundary conditions, Probab. Theory Related Fields 82 (1989) 489-526. MR 91a:60154
28. D. Ocone and E. Pardoux, A generalized Itô-Ventzell formula. Application to a class of anticipating stochastic differential equations, Ann. Inst. H. Poincaré Probab. Statist. 25 (1989) 39-71. MR 90g:60055
29. G. Papanicolaou, Waves in one dimensional random media, Lecture Notes in Mathematics vol. 1362, Ecole d'Été de Probabilités de Saint-Flour XXV (1985-87), Springer, Berlin, 1988, 205-276. MR 90j:82004
30. E. Pardoux, Equations du lissage nonlineaire, Lect. Notes Control Inf. Sci. vol. 61, Filtering and Control of Random Processes, H. Korezlioglu, G. Mazziotto and J. Szpirglas, eds., Springer, Berlin, 1986, 206-218. MR 87i:93104
31. D. Talay and L. Tubaro, Expansion of the global error for numerical Schemes solving stochastic differential equations, Stochastic Anal. Appl. 8 (1990) 483-510. MR 92e:60124
32. O. Zeitouni and A. Dembo, A maximum a posteriori estimator for trajectories of diffusions, Stochastics 20 (1987) 221-246 (Erratum) 20 (1987) 341. MR 88b:93050a,b
33. O. Zeitouni and A. Dembo, An existence theorem for maximum a posteriori estimation of trajectories of diffusions, Stochastics 23 (1988) 197-218. MR 89g:93100
34. O. Zeitouni and A. Dembo, A change of variables formula for Stratonovich integrals and existence of solutions for two-point stochastic boundary value problems, Probab. Theory Related Fields 84 (1990) 411-425. MR 91h:60061