Pathwise Integration and functional calculus for paths with finite quadratic variation

Abstract

This thesis develops a pathwise calculus for non-anticipative functionals of paths with finite quadratic variation and studies its relation with the theory of controlled paths.

We study the mathematical properties of a pathwise integral defined as a limit of Riemann sums for a class of non-anticipative gradient-type integrands. We establish for this integral a pathwise isometry property, analogous to the well-known Ito isometry for stochastic integrals, and obtain a pathwise 'signal plus noise' decomposition as a unique sum of a pathwise integral and a component with zero quadratic variation for regular functionals of an irregular path with non-vanishing quadratic variation. Our results are strictly pathwise but apply to typical paths of continuous semimartingales.

In the second part of the thesis we explore the relations between this non-anticipative functional calculus and the theory of controlled paths. We show that a regular functional generates a family of controlled paths whose `Gubinelli derivative' may be represented as a directional derivative. Conversely, we show that a family of controlled paths parameterized by the underlying control function may be represented as a vertically differentiable functional. This result leads to a chain rule for controlled paths and systematic way of constructing them.

In the last part of the thesis we extend these results to functionals of discontinuous paths which are right-continuous with left limits.