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 wellposedness result for viscosity solutions of PPDEs when the underlying space is a separable Hilbert space. We also observe that, in contrast with the finitedimensional 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
Gozzi F.;Rosestolato M.;
2018-01-01
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 wellposedness result for viscosity solutions of PPDEs when the underlying space is a separable Hilbert space. We also observe that, in contrast with the finitedimensional 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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
