'Scuola Normale Superiore - Edizioni della Normale'
Abstract
This thesis is devoted to study delay/path-dependent stochastic differential equations
and their connection with partial differential equations in infinite dimensional spaces,
possibly path-dependent. We address mathematical problems arising in hedging a derivative
product for which the volatility of the underlying assets as well as the claim may
depend on the past history of the assets themselves.
The starting point is to provide a robust framework for working with mild solutions
to path-dependent SDEs: well-posedness, continuity with respect to the data, regularity
with respect to the initial condition. This is done in Chapter 1. In Chapter 2, under
Lipschitz conditions on the data, we prove the directional regularity needed in order to
write the hedging strategy. In Chapter 3 we introduce a new notion of viscosity solution
to semilinear path-dependent PDEs in Hilbert spaces (PPDEs), we prove well-posedness
and show that the solution is given by the Fyenman-Kac formula. In Chapter 4 we
extend to Hilbert spaces the functional It\uafo calculus and, under smooth assumptions on
the data, we prove a path-dependent It\uafo\u2019s formula, show existence of classical solutions
to PPDEs, and obtain a Clark-Ocone type formula. In Chapter 5 we introduce a new
notion of C0-semigroup suitable to be applied to Markov transition semigroups, hence
to mild solutions to Kolmogorov PDEs, and we prove all the basic results analogous
to those available for C0-semigroups in Banach spaces. Additional theoretical results
for stochastic analysis in Hilbert spaces, regarding stochastic convolutions, are given in
Appendix A.
Our methodology varies among different chapters. Path-dependent models can be
studied in their original path-dependent form or by representing them as non-pathdependent
models in infinite dimension. We exploit both approaches. We treat pathdependent
Kolmogorov equations in infinite dimension with two notions of solution: classical
and viscosity solutions. Each approach leads to original results in each chapter
