Roughness properties of paths and signals
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

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 variation, then expand results to the more general setting of p-th order variation.
The first chapter discusses some motivations and background for the questions explored in the thesis and provides an overview of the results.
In the second chapter, we study quadratic variation along a sequence of partitions and its dependence with respect to the choice of the partition sequence. We introduce a property which we call quadratic roughness, and show that for H ̈older-continuous paths satisfying this roughness condition, the quadratic variation along ‘balanced’ partitions is invariant with respect to the choice of the partition sequence. Typical paths of Brownian motion satisfy this quadratic roughness property almost-surely along partitions with fine enough mesh. Using these results we derive a formulation of the pathwise F ̈ollmer-Itˆo calculus which is invariant with respect to the partition sequences. Furthermore, we provide an invariance result for local time under quadratic roughness.
In the third chapter, instead of balanced partition sequences (which is a key condition in Chapter 2) we consider (finitely) refining partition sequences, without any bound on mesh size. We construct a generalized Haar basis along any such finite refining sequence of partitions. We provide a closed-form representation of quadratic variation in terms of Faber-Schauder coefficients along this basis. Further, we construct a class of continuous processes with linear and prescribed quadratic variations along any given finitely refining partition sequence. We provide an example of a rough class of continuous processes with invariant quadratic variations along finitely refining sequences of partitions. Brownian motion belongs to this ‘rough’ class, but we also give examples of processes with 1/2 -H ̈older continuity in this class. Finally, we extend these constructions to higher dimensions.
In the fourth chapter of the thesis, we consider a more general concept of roughness based on p-th variation and the associated notions of variation and roughness index of a continuous function. We define the normalized p-th variation of a path and use it to introduce a pathwise estimator to estimate the order of roughness of a signal. We investigate the finite sample performance of our estimator for measuring the roughness of sample paths of stochastic processes using detailed numerical experiments based on sample paths of fractional Brownian motion and Takagi-Landsberg functions. In the final chapter we use our ‘roughness’ estimator (discussed in Chapter 4) to investigate the statistical evidence for the use of ‘rough’ fractional processes with Hurst exponent H < 0.5 for the modelling of volatility of financial assets, using a non-parametric, model-free approach. Detailed numerical experiments based on stochastic volatility models show that, even when the instantaneous volatility has diffusive dynamics with the same roughness as Brownian motion, the realized volatility exhibits rough behaviour corresponding to a Hurst exponent significantly smaller than 0.5, which suggests that the origin of the roughness observed in realized volatility time-series lies in the estimation error rather than the volatility process itself. Comparison of roughness estimates for realized and instantaneous volatility in fractional volatility models with different values of Hurst exponent shows that, irrespective of the value of H, realized volatility always exhibits ‘rough’ behaviour with an apparent Hurst index ˆH < 0.5 but this is not necessarily indicative of a similar rough behaviour of the spot volatility process which may have H ≥ 1/2.