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Title: Markovian rough paths
Author: Ogrodnik, Marcel Bogdan
ISNI:       0000 0004 6423 6610
Awarding Body: Imperial College London
Current Institution: Imperial College London
Date of Award: 2016
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The accumulated local p-variation functional, originally presented by Cass et al. (2013), arises naturally in the theory of rough paths in estimates both for solutions to rough differential equations (RDEs), and for the higher-order terms of the signature (or Lyons lift). In stochastic examples, it has been observed that the tails of the accumulated local p-variation functional typically decay much faster than the tails of classical p-variation. This observation has been decisive, e.g. for problems involving Malliavin calculus for Gaussian rough paths as illustrated in the work by Cass et al. (2015). All of the examples treated so far have been in this Gaussian setting, that contains a great deal of additional structure. In this paper we work in the context of Markov processes on a locally compact Polish space E, which are associated to a class of Dirichlet forms. In this general framework, we first prove a better-than-exponential tail estimate for the accumulated local p-variation functional derived from the intrinsic metric of this Dirichlet form. By then specialising to a class of Dirichlet forms on the step-⌊p⌋ free nilpotent group, which are subelliptic in the sense of Fefferman-Phong, we derive a better than exponential tail estimate for a class of Markovian rough paths. This class includes, but also goes beyond, the examples studied by Friz and Victoir (2008). We comment on the significance of these estimates to recent results, including the results of Hao (2014) and Chevyrev and Lyons (2015).
Supervisor: Cass, Thomas ; Crisan, Dan Sponsor: Imperial College London
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral