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Path-dependent equations and viscosity solutions in infinite dimension

Abstract : Path Dependent PDE's (PPDE's) are natural objects to study when one deals with non Markovian models. Recently, after the introduction (see [12]) of the so-called pathwise (or functional or Dupire) calculus, various papers have been devoted to study the well-posedness of such kind of equations, both from the point of view of regular solutions (see e.g. [18]) and viscosity solutions (see e.g. [13]), in the case of finite dimensional underlying space. In this paper, motivated by the study of models driven by path dependent stochastic PDE's, we give a first well-posedness result for viscosity solutions of PPDE's when the underlying space is an infinite dimensional Hilbert space. The proof requires a substantial modification of the approach followed in the finite dimensional case. We also observe that, differently from the finite dimensional case, our well-posedness result, even in the Markovian case, apply to equations which cannot be treated, up to now, with the known theory of viscosity solutions.
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Contributor : Andrea Cosso <>
Submitted on : Tuesday, February 17, 2015 - 3:42:40 PM
Last modification on : Friday, March 27, 2020 - 3:56:26 AM
Document(s) archivé(s) le : Monday, May 18, 2015 - 10:40:18 AM


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  • HAL Id : hal-01117693, version 1
  • ARXIV : 1502.05648


Andrea Cosso, Salvatore Federico, Fausto Gozzi, Mauro Rosestolato, Nizar Touzi. Path-dependent equations and viscosity solutions in infinite dimension. 2015. ⟨hal-01117693⟩



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