Path Dependent PDE's (PPDE's) are natural objects to study when one deals with non Markovian models. Recently, after the introduction (see [12]) of the so-called pathwise (or functional or Dupire) calculus, various papers have been devoted to study the well-posedness of such kind of equations, both from the point of view of regular solutions (see e.g. [18]) and viscosity solutions (see e.g. [13]), in the case of finite dimensional underlying space. In this paper, motivated by the study of models driven by path dependent stochastic PDE's, we give a first well-posedness result for viscosity solutions of PPDE's when the underlying space is an infinite dimensional Hilbert space. The proof requires a substantial modification of the approach followed in the finite dimensional case. We also observe that, differently from the finite dimensional case, our well-posedness result, even in the Markovian case, apply 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 / Cosso, Andrea; Federico, Salvatore; Gozzi, Fausto; Rosestolato, Mauro; Touzi, Nizar. - In: ANNALS OF PROBABILITY. - ISSN 0091-1798. - 46:1(2018), pp. 126-174. [10.1214/17-AOP1181]

Path-dependent equations and viscosity solutions in infinite dimension

GOZZI, FAUSTO;ROSESTOLATO, MAURO;
2018

Abstract

Path Dependent PDE's (PPDE's) are natural objects to study when one deals with non Markovian models. Recently, after the introduction (see [12]) of the so-called pathwise (or functional or Dupire) calculus, various papers have been devoted to study the well-posedness of such kind of equations, both from the point of view of regular solutions (see e.g. [18]) and viscosity solutions (see e.g. [13]), in the case of finite dimensional underlying space. In this paper, motivated by the study of models driven by path dependent stochastic PDE's, we give a first well-posedness result for viscosity solutions of PPDE's when the underlying space is an infinite dimensional Hilbert space. The proof requires a substantial modification of the approach followed in the finite dimensional case. We also observe that, differently from the finite dimensional case, our well-posedness result, even in the Markovian case, apply 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 / Cosso, Andrea; Federico, Salvatore; Gozzi, Fausto; Rosestolato, Mauro; Touzi, Nizar. - In: ANNALS OF PROBABILITY. - ISSN 0091-1798. - 46:1(2018), pp. 126-174. [10.1214/17-AOP1181]
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11385/147259
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