PATH-DEPENDENT EQUATIONS AND VISCOSITY SOLUTIONS IN INFINITE DIMENSION

• Published in: The Annals of probability 2018-01, Vol.46 (1), p.126-174
• Main Authors: Cosso, Andrea, Federico, Salvatore, Gozzi, Fausto, Rosestolato, Mauro, Touzi, Nizar
• Format: Article
• Language: English
• Subjects: Calculus; Hilbert space; Markov chains; Nonlinear equations; Partial differential equations; Stochastic models; Studies; Viscosity; Well posed problems
• ISSN: 0091-1798; 2168-894X
• DOI: 10.1214/17-AOP1181

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.