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Mathematics  2013 

Jain-Monrad criterion for rough paths and applications

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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-Monrad [Ann. Probab. (1983)]. 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 It\^o-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 H\"ormander theory. The results include and extend those of [arXiv:1211.0046].


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