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INFINITE DIMENSIONAL METHODS FOR PATH-DEPENDENT SDEs: KOLMOGOROV EQUATIONS AND ITO FORMULAE
Giovanni Zanco (Universitą di Pisa)
Sala conferenze IMATI-CNR - Venerdì 29 Maggio 2015 h.15:00
Abstract. Path-dependent stochastic differential equations are (non-markovian) equations whose coefficients are allowed to depend on the whole trajectory of the solution up to the present time, and are a powerful tool in modeling complex evolution systems with memory that appear in finance, engineering and biology.
Even when the state space is finite dimensional, they are intrinsecally infinite dimensional. I will show how path-dependent SDEs can be studied in a product space framework (inspired by the theory for delay equations) using standard differential and topological structures. This framework is helpful to prove existence and uniqueness of classical solutions to path-dependent Kolmogorov-type PDEs on the space of continuous paths, to obtain probabilistic representation formulas for such solutions and moreover to prove Ito-type formulae for functionals of paths of continuous semimartingales, thus providing a counterpart of the functional Ito calculus developed by Dupire, Cont and Fournié.
It also provides an insight on the role and the analytical structure of the so-called horizontal derivative, which is a key object in the study of path-dependent equations.
The results I will present have been obtained in collaboration with Franco Flandoli (Unipi) and Francesco Russo (Ensta-ParisTech).
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