Infinite-dimensional methods for path-dependent stochastic differential equations

This thesis addresses the problem of extending results of stochastic analysis from the classical Markovian setting to the path-dependent setting. The two main aspects considered are: the relation between path-dependent stochastic differential equations and partial differential equations and change of variable formulae of Ito type. It also provides a comparison between the methods developed and the Functional Ito calculus recently introduced and studied by other authors. The main results obtained in the thesis are: existence and uniqueness of classical solutions to an infinite-dimensional Kolmogorov equation associated to a path-dependent stochastic differential equation; existence of Ito formulae in Hilbert and Banach spaces in situations where some of the terms appearing are ill-defined; a comparison result between the infinite-dimensional method presented and the Functional Ito calculus; a proof of existence and uniqueness of classical solutions to the path-dependent Kolmogorov equation based on the previous results. The methods used are based on a reformulation of the problems in infinite-dimensional Hilbert and Banach spaces; in particular, tools of stochastic analysis in space of continuous and càdlàg functions are investigated. The results obtained about Komogorov equations exhibit significative differences from their analogue in the classical case, both from the point of view of necessary conditions and of regularising properties of the equations. The Ito formulae obtained are suitable to be applied to different contexts and are therefore investigated in a general abstract setting. Beside their application to path-dependent equations, an application to stochastic equations with group generators is presented