Markovian rough paths
File(s)
Author(s)
Ogrodnik, Marcel Bogdan
Type
Thesis
Abstract
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).
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).
Version
Open Access
Date Issued
2016-12
Date Awarded
2017-03
Copyright Statement
Attribution NoDerivatives 4.0 International Licence (CC BY-ND)
Advisor
Cass, Thomas
Crisan, Dan
Sponsor
Imperial College London
Publisher Department
Mathematics
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)