Path-dependent equations and viscosity solutions in infinite dimension

Andrea Cosso, Salvatore Federico, Fausto Gozzi, Mauro Rosestolato, Nizar Touzi

Research output: Contribution to journalArticlepeer-review


Path-dependent partial differential equations (PPDEs) are natural objects to study when one deals with non-Markovian models. Recently, after the introduction of the so-called pathwise (or functional or Dupire) calculus [see Dupire (2009)], in the case of finite-dimensional underlying space various papers have been devoted to studying the well-posedness of such kind of equations, both from the point of view of regular solutions [see, e.g., Dupire (2009) and Cont (2016) Stochastic Integration by Parts and Functional Itô Calculus 115-207, Birkhäuser] and viscosity solutions [see, e.g., Ekren et al. (2014) Ann. Probab. 42 204-236]. In this paper, motivated by the study of models driven by path-dependent stochastic PDEs, we give a first wellposedness result for viscosity solutions of PPDEs when the underlying space is a separable Hilbert space. We also observe that, in contrast with the finitedimensional case, our well-posedness result, even in the Markovian case, applies to equations which cannot be treated, up to now, with the known theory of viscosity solutions.

Original languageEnglish (US)
Pages (from-to)126-174
Number of pages49
JournalAnnals of Probability
Issue number1
StatePublished - Jan 2018


  • Partial differential equations in infinite dimension
  • Path-dependent partial differential equations
  • Path-dependent stochastic differential equations
  • Viscosity solutions

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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