The Jain–Monrad criterion for rough paths and applications to random Fourier series and non-Markovian Hörmander theory

By Peter K. Friz, Benjamin Gess, Archil Gulisashvili. 2013

62 citations recorded by the source corpus.

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.

Subfields: Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/effaceba078844eef05603d165e079792ae64387

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In Finance

  1. Samy Tindel
  2. Cheng Ouyang
  3. E. Nualart
  4. Fabrice Baudoin
  5. Peter K. Friz
  6. Alexander Schied
  7. Martin Hairer
  8. David Nualart

In Mathematical Physics

  1. E. Nualart
  2. Fabrice Baudoin
  3. Samy Tindel
  4. Alexander Schied
  5. R. Chetrite
  6. Hugo Touchette

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