Path-dependent equations and viscosity solutions in infinite dimension

• Andrea Cosso ^[1] ; Salvatore Federico ^[2] ; Fausto Gozzi ^[3] ; Mauro Rosestolato ^[4] ; Nizar Touzi ^[4]
  1. [1] Polytechnic University of Milan

     Polytechnic University of Milan

     Milán, Italia

     
  2. [2] Università di Siena (Italy)
  3. [3] Libera Università Internazionale degli Studi Sociali Guido Carli( nLUISS) University (Italy)
  4. [4] Ecole Polytechnique (France)

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• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 46, Nº. 1, 2018, págs. 126-174
• Idioma: inglés
• DOI: 10.1214/17-AOP1181
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• Resumen

  • 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 well-posedness result for viscosity solutions of PPDEs when the underlying space is a separable Hilbert space. We also observe that, in contrast with the finite-dimensional 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.