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Weak approximations. A Malliavin calculus approach


Author: Arturo Kohatsu-Higa
Journal: Math. Comp. 70 (2001), 135-172
MSC (2000): Primary 60H07, 60H35, 65C30, 34B99
DOI: https://doi.org/10.1090/S0025-5718-00-01201-1
Published electronically: March 2, 2000
MathSciNet review: 1680895
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Abstract:

We introduce a variation of the proof for weak approximations that is suitable for studying the densities of stochastic processes which are evaluations of the flow generated by a stochastic differential equation on a random variable that may be anticipating. Our main assumption is that the process and the initial random variable have to be smooth in the Malliavin sense. Furthermore, if the inverse of the Malliavin covariance matrix associated with the process under consideration is sufficiently integrable, then approximations for densities and distributions can also be achieved. We apply these ideas to the case of stochastic differential equations with boundary conditions and the composition of two diffusions.


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Additional Information

Arturo Kohatsu-Higa
Affiliation: Universitat Pompeu Fabra. Departament d’Economia. Ramón Trias Fargas 25-27. 08005 Barcelona. Spain
Email: kohatsu@upf.es

DOI: https://doi.org/10.1090/S0025-5718-00-01201-1
Keywords: Stochastic differential equations, boundary conditions, weak approximation, numerical analysis
Received by editor(s): June 9, 1998
Received by editor(s) in revised form: March 2, 1999
Published electronically: March 2, 2000
Additional Notes: This article was partially written while the author was visiting the Department of Mathematics at Kyoto University with a JSPS fellowship. His research was partially supported by a DGES grant.
Article copyright: © Copyright 2000 by Arturo Kohatsu-Higa

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