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January 2016 The Jain–Monrad criterion for rough paths and applications to random Fourier series and non-Markovian Hörmander theory
Peter K. Friz, Benjamin Gess, Archil Gulisashvili, Sebastian Riedel
Ann. Probab. 44(1): 684-738 (January 2016). DOI: 10.1214/14-AOP986

Abstract

We discuss stochastic calculus for large classes of Gaussian processes, based on rough path analysis. Our key condition is a covariance measure structure combined with a classical criterion due to Jain and Monrad [Ann. Probab. 11 (1983) 46–57]. This condition is verified in many examples, even in absence of explicit expressions for the covariance or Volterra kernels. Of special interest are random Fourier series, with covariance given as Fourier series itself, and we formulate conditions directly in terms of the Fourier coefficients. We also establish convergence and rates of convergence in rough path metrics of approximations to such random Fourier series. An application to SPDE is given. Our criterion also leads to an embedding result for Cameron–Martin paths and complementary Young regularity (CYR) of the Cameron–Martin space and Gaussian sample paths. CYR is known to imply Malliavin regularity and also Itô-like probabilistic estimates for stochastic integrals (resp., stochastic differential equations) despite their (rough) pathwise construction. At last, we give an application in the context of non-Markovian Hörmander theory.

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Peter K. Friz. Benjamin Gess. Archil Gulisashvili. Sebastian Riedel. "The Jain–Monrad criterion for rough paths and applications to random Fourier series and non-Markovian Hörmander theory." Ann. Probab. 44 (1) 684 - 738, January 2016. https://doi.org/10.1214/14-AOP986

Information

Received: 1 November 2013; Revised: 1 July 2014; Published: January 2016
First available in Project Euclid: 2 February 2016

zbMATH: 1347.60097
MathSciNet: MR3456349
Digital Object Identifier: 10.1214/14-AOP986

Subjects:
Primary: 60G15, 60H15
Secondary: 42A32, 60G17

Rights: Copyright © 2016 Institute of Mathematical Statistics

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Vol.44 • No. 1 • January 2016
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