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A Lévy area between Brownian motion and rough paths with applications to robust nonlinear filtering and rough partial differential equations

Author

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  • Diehl, Joscha
  • Oberhauser, Harald
  • Riedel, Sebastian

Abstract

We give meaning to differential equations with a rough path term and a Brownian noise term and study their regularity, that is we are interested in equations of the type Stη=S0+∫0ta(Srη)dr+∫0tb(Srη)∘dBr+∫0tc(Srη)dηr where η is a deterministic geometric, step-2 rough path and B is a multi-dimensional Brownian motion. We then give two applications: a Feynman–Kac formula for RPDEs and a robust version of the conditional expectation that appears in the nonlinear filtering problem. En passant, we revisit the recent integrability estimates of Cass et al. (2013) for rough differential equations with Gaussian driving signals which might be of independent interest.

Suggested Citation

  • Diehl, Joscha & Oberhauser, Harald & Riedel, Sebastian, 2015. "A Lévy area between Brownian motion and rough paths with applications to robust nonlinear filtering and rough partial differential equations," Stochastic Processes and their Applications, Elsevier, vol. 125(1), pages 161-181.
    • Handle: RePEc:eee:spapps:v:125:y:2015:i:1:p:161-181
    DOI: 10.1016/j.spa.2014.08.005
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    Cited by:

    1. Hocquet, Antoine & Nilssen, Torstein & Stannat, Wilhelm, 2020. "Generalized Burgers equation with rough transport noise," Stochastic Processes and their Applications, Elsevier, vol. 130(4), pages 2159-2184.
    2. Coghi, Michele & Nilssen, Torstein, 2021. "Rough nonlocal diffusions," Stochastic Processes and their Applications, Elsevier, vol. 141(C), pages 1-56.

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