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Approximations of the brownian rough path with applications to stochastic analysis

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  • Mathematics


doi:10.1016/j.anihpb.2004.05.003 Ann. I. H. Poincaré – PR 41 (2005) 703–724 A Abstra A g natura This Motion As a ian Mo Motion Proces (1998) simpli modul  2004 Résum Un Le gro Cett brown Solé) e cela gé [Rev. M les top * Cor E-m 0246-0 doi:10. pproximations of the Brownian rough path with applications to stochastic analysis Peter Friz a, Nicolas Victoir b,∗ a 251 Mercer St., New York, NY 10012, USA b Magdalen College, Oxford OX1 4AU, UK Received 25 August 2003; received in revised form 15 March 2004; accepted 11 May 2004 Available online 11 September 2004 ct eometric p-rough path can be seen to be a genuine path of finite p-variation with values in a Lie group equipped with a l distance. The group and its distance lift (Rd ,+,0) and its Euclidean distance. approach allows us to easily get a precise modulus of continuity for the Enhanced Brownian Motion (the Brownian and its Lévy Area). first application, extending an idea due to Millet and Sanz-Solé, we characterize the support of the Enhanced Brown- tion (without relying on correlation inequalities). Secondly, we prove Schilder’s theorem for this Enhanced Brownian . As all results apply in Hölder (and stronger) topologies, this extends recent work by Ledoux, Qian, Zhang [Stochastic s. Appl. 102 (2) (2002) 265–283]. Lyons’ fine estimates in terms of control functions [Rev. Mat. Iberoamericana 14 (2) 215–310] allow us to show that the Itô map is still continuous in the topologies we introduced. This provides new and fied proofs of the Stroock–Varadhan support theorem and the Freidlin–Wentzell theory. It also provides a short proof of us of continuity for diffusion processes along old results by Baldi. Elsevier SAS. All rights reserved. é p-rough path est un chemin de p-variation finie à valeurs dans un groupe de Lie muni d’une distance sous-riemannienne. upe et sa distance géneralisent (Rd ,+,0) et la distance euclidienne. e approche nous permet d’obtenir un module de conti

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