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Approximations of the Brownian rough path with applications to stochastic analysis. (English) Zbl 1080.60021
For a fixed real number \(p\in]2;3[\), a new presentation of the set \(G\Omega_p(\mathbb{R}^d)\) of geometric \(p\)-rough paths is proposed; denoting by \(G\) the free nilpotent group of step \(2\) over \(\mathbb{R}^d\), the authors equip \((G;\otimes)\) with a homogeneous, sub-additive norm, and characterise \(G\Omega_p(\mathbb{R}^d)\) as the closure of the set of all \(G\)-valued “smooth paths” under the \(p\)-variation metric. Firstly, this enables them to give a precise estimation for the modulus of continuity of the enhanced Brownian motion (EBM). Several topologies on \(G\Omega_p(\mathbb{R}^d)\) are considered, corresponding e.g. to “modulus of continuity” type norms, and the continuity of the Itô map is established for these finer topologies. The authors also establish the convergence (respective to these topologies) of some sequences of smooth rough paths to the EBM, using approximations obtained e.g. by conditioning along dyadic filtrations. A support theorem is then established for the law of the EBM, and Schilder’s theorem for the EBM is also obtained as a consequence of the continuity of the Itô map; this improves earlier results of M. Ledoux, Z. Qian and T. Zhang [Stochastic Processes Appl. 102, No. 2, 265–283 (2002; Zbl 1075.60510)], in that the topologies under consideration are finer.

MSC:
60F10 Large deviations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60L20 Rough paths
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