Roughness properties of paths and signals

Thesis

Functions and processes with irregular behaviour in time are ubiquitous in physics, engineering, and finance and have been the focus of various pathwise theories of integration in stochastic analysis, in which the degree of 'roughness' of the function plays an important role. This thesis focuses on various concepts of 'roughness' for continuous functions and processes and their interplay with pathwise integration. We first explore these issues using the concept of pathwise quadratic variat...

Expand abstract

Authors: Das, P

• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: Quadratic roughness; Normalized p-th variation; Rough volatility; p-th variation; Schauder basis; Haar basis; Quadratic variation; Brownian motion; Spot and realized volatility; Takagi class of functions; Fractional Brownian motion; Pathwise methods; F ̈ollmer-Itˆo map; Roughness estimator
• Subjects: Stochastic integrals; Mathematics; Functions of several real variables; Calculus of variations; Analysis; Schauder bases