The Annals of Probability

Path-dependent equations and viscosity solutions in infinite dimension

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

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Abstract

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.

Article information

Source
Ann. Probab., Volume 46, Number 1 (2018), 126-174.

Dates
Received: February 2015
Revised: October 2016
First available in Project Euclid: 5 February 2018

Permanent link to this document
https://projecteuclid.org/euclid.aop/1517821220

Digital Object Identifier
doi:10.1214/17-AOP1181

Mathematical Reviews number (MathSciNet)
MR3758728

Zentralblatt MATH identifier
06865120

Subjects
Primary: 35D40: Viscosity solutions 35R15: Partial differential equations on infinite-dimensional (e.g. function) spaces (= PDE in infinitely many variables) [See also 46Gxx, 58D25] 60H15: Stochastic partial differential equations [See also 35R60] 60H30: Applications of stochastic analysis (to PDE, etc.)

Keywords
Viscosity solutions path-dependent stochastic differential equations path-dependent partial differential equations partial differential equations in infinite dimension

Citation

Cosso, Andrea; Federico, Salvatore; Gozzi, Fausto; Rosestolato, Mauro; Touzi, Nizar. Path-dependent equations and viscosity solutions in infinite dimension. Ann. Probab. 46 (2018), no. 1, 126--174. doi:10.1214/17-AOP1181. https://projecteuclid.org/euclid.aop/1517821220


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