Open Access
July 2017 Tail estimates for Markovian rough paths
Thomas Cass, Marcel Ogrodnik
Ann. Probab. 45(4): 2477-2504 (July 2017). DOI: 10.1214/16-AOP1117


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 $\lfloor p\rfloor $ 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].


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Thomas Cass. Marcel Ogrodnik. "Tail estimates for Markovian rough paths." Ann. Probab. 45 (4) 2477 - 2504, July 2017.


Received: 1 December 2014; Revised: 1 September 2015; Published: July 2017
First available in Project Euclid: 11 August 2017

zbMATH: 06786086
MathSciNet: MR3693967
Digital Object Identifier: 10.1214/16-AOP1117

Primary: 47D07 , 60B15 , 60G20 , 65C30

Keywords: Dirichlet form , Markov process , rough path theory , tail estimates

Rights: Copyright © 2017 Institute of Mathematical Statistics

Vol.45 • No. 4 • July 2017
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