THE JAIN-MONRAD CRITERION FOR ROUGH PATHS AND APPLICATIONS TO RANDOM FOURIER SERIES AND NON-MARKOVIAN HORMANDER THEORY

• 成果类型: Article
• 署名作者: Friz, Peter K.; Gess, Benjamin; Gulisashvili, Archil; Riedel, Sebastian
• 署名单位: Technical University of Berlin; Humboldt University of Berlin; Leibniz Association; Weierstrass Institute for Applied Analysis & Stochastics; University of Chicago; University System of Ohio; Ohio University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/14-AOP986
• 发表日期: 2016
• 页码: 684-738
• 关键词: differential-equations driven; Integrability; spdes

摘要

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 Ito-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 Hormander theory.