Topics in stochastic calculus in infinite dimension for financial applications

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¯o calculus and, under smooth assumptions on the data, we prove a path-dependent It¯o’s 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