Tail estimates for Markovian rough paths
File(s)
Published version
Author(s)
Cass, T
Ogrodnik, M
Type
Journal Article
Abstract
The accumulated local p-variation functional [Ann. Probab. 41 (213) 3026–3050] 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, for example, for problems involving Malliavin calculus for Gaussian rough paths [Ann. Probab. 43 (2015) 188–239].
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 sub-elliptic in the sense of Fefferman–Phong, we derive a better than exponential tail estimate for a class of Markovian rough paths. This class includes the examples studied in [Probab. Theory Related Fields 142 (2008) 475–523]. We comment on the significance of these estimates to recent papers, including the results of Ni Hao [Personal communication (2014)] and Chevyrev and Lyons [Ann. Probab. To appear].
-variation. This observation has been decisive, for example, for problems involving Malliavin calculus for Gaussian rough paths [Ann. Probab. 43 (2015) 188–239].
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 sub-elliptic in the sense of Fefferman–Phong, we derive a better than exponential tail estimate for a class of Markovian rough paths. This class includes the examples studied in [Probab. Theory Related Fields 142 (2008) 475–523]. We comment on the significance of these estimates to recent papers, including the results of Ni Hao [Personal communication (2014)] and Chevyrev and Lyons [Ann. Probab. To appear].
Date Issued
2017-07-01
Date Acceptance
2016-04-11
Citation
Annals of Probability, 2017, 45 (4), pp.2477-2504
ISSN
0091-1798
Publisher
Institute of Mathematical Statistics (IMS)
Start Page
2477
End Page
2504
Journal / Book Title
Annals of Probability
Volume
45
Issue
4
Copyright Statement
© Institute of Mathematical Statistics, 2017
Sponsor
Engineering & Physical Science Research Council (EPSRC)
Grant Number
EP/M00516X/1
Subjects
math.PR
math.PR
Publication Status
Published
Date Publish Online
2017-08-11