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Title: Pathwise integration and functional calculus for paths with finite quadratic variation
Author: Ananova, Anna
ISNI:       0000 0004 7658 855X
Awarding Body: Imperial College London
Current Institution: Imperial College London
Date of Award: 2019
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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.
Supervisor: Cont, Rama Sponsor: Imperial College London
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral