WIAS Preprint No. 2506, (2018)

Solving linear parabolic rough partial differential equations



Authors

  • Bayer, Christian
    ORCID: 0000-0002-9116-0039
  • Belomestny, Denis
  • Redmann, Martin
    ORCID: 0000-0001-5182-9773
  • Riedel, Sebastian
  • Schoenmakers, John G. M.
    ORCID: 0000-0002-4389-8266

2010 Mathematics Subject Classification

  • 65C30 65C05 60H15

Keywords

  • Rough paths, rough partial differential equations, Feynman-Kac formula, regression

DOI

10.20347/WIAS.PREPRINT.2506

Abstract

We study linear rough partial differential equations in the setting of [Friz and Hairer, Springer, 2014, Chapter 12]. More precisely, we consider a linear parabolic partial differential equation driven by a deterministic rough path W of Hölder regularity α with ⅓ < α ≤ ½ . Based on a stochastic representation of the solution of the rough partial differential equation, we propose a regression Monte Carlo algorithm for spatio-temporal approximation of the solution. We provide a full convergence analysis of the proposed approximation method which essentially relies on the new bounds for the higher order derivatives of the solution in space. Finally, a comprehensive simulation study showing the applicability of the proposed algorithm is presented.

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