Path-dependent PDEs (PPDEs) are natural objects to study when one deals with non Markovian models. Recently, after the intro- duction of the so-called pathwise (or functional or Dupire) calculus (see [15]), 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. [15, 9]) and viscosity solutions (see e.g. [16]). 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.

Path-dependent equations and viscosity solutions in infinite dimension / A. Cosso, S. Federico, F. Gozzi, M. Rosestolato, N. Touzi. - In: ANNALS OF PROBABILITY. - ISSN 0091-1798. - 46:1(2018 Jan), pp. 126-174. [10.1214/17-AOP1181]

Path-dependent equations and viscosity solutions in infinite dimension

A. Cosso
Primo
;
2018-01

Abstract

Path-dependent PDEs (PPDEs) are natural objects to study when one deals with non Markovian models. Recently, after the intro- duction of the so-called pathwise (or functional or Dupire) calculus (see [15]), 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. [15, 9]) and viscosity solutions (see e.g. [16]). 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.
viscosity solutions; path-dependent stochastic differential equations; path-dependent partial differential equations; partial differential equations in infinite dimension
Settore MAT/06 - Probabilita' e Statistica Matematica
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/2434/931972
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