Functional calculus for cadlag paths and applications to model-free finance

Abstract

This thesis synthesise my research on analysis and control of path-dependent random systems under uncertainty. In the first chapter, we revisit Foellmer's concept of pathwise quadratic variation for a cadlag path and show that his definition can be reformulated in terms of convergence of quadratic sums in the Skorokhod topology. This new definition is simpler and amenable to define higher order variation for a cadlag path.

In the second chapter, we introduced a new topology for functionals and adopted an abstract formulation of Functional calculus on generic domain based on the differentials introduced by Dupire (2009), Cont & Fournie (2010). Our aim is not to generalise an existing rich theory for irregular paths e.g. Lyons (1998), Friz & Hairer (2014) but to introduce a bespoke and yet versatile calculus for causal random system in general and mathematical finance in particular, in order to solve problems practically as well as bring in new aspects under uncertainty.

In the final chapter, we apply functional calculus to study mathematical finance under uncertainty. We first show that every self-financing portfolio can be represented by a pathwise integral and that every generic market is arbitrage free, a fundamental property that is linked to the solution, which is characterised by a fully non-linear path dependent equation, to the optimal hedging problem under uncertainty. In particular, we obtain explicit solution for the Asian option.