Fractional Brownian Motion: Theory and Financial Applications

Below is my Final Year Mathematics dissertation.

Natural phenomena often exhibit persistence patterns that standard stochastic models fail to capture. A river’s flood levels tend to cluster over decades; financial volatility shows "memory" across time scales; network traffic displays self-similar bursts. These observations challenge the Markovian paradigm underlying classical stochastic calculus.

Mandelbrot and Van Ness (1968) introduced fractional Brownian motion as a natural extension of standard Brownian motion, incorporating memory effects through the Hurst parameter H. This parameter quantifies the strength and direction of temporal dependencies, with H > 1/2 corresponding to persistence (trends tend to continue), H < 1/2 to anti-persistence (trends tend to reverse), and H=1/2 recovering standard Brownian motion with independent increments. This dissertation presents a comprehensive mathematical analysis of fractional Brownian motion (fBm), developing both its theoretical foundations and practical applications with particular emphasis on financial models. The dissertation is structured as follows:

• Chapter 1 establishes the measure-theoretic framework and introduces the standard Brownian motion through its covariance structure, establishing the fundamental difference in structure that motivates the introduction of its fractional extension.

• Chapter 2 introduces fractional Brownian motion, emphasising its generalisation of standard Brownian motion through the introduction of the Hurst parameter. We examine its memory structure through time and spectral perspectives, and analyses path properties, proving that fBm exhibits (𝐻 − 𝜀)-Hölder continuity and a phase transition in quadratic variation at 𝐻 = 1 2

• Chapter 3 develops integration theory for fBm motivated through financial applications, proving that standard Itô calculus fails when 𝐻 ≠ 1 2 due to the non semimartingale property. For 𝐻 > 1 2, we establish Pathwise integration theory based on Young’s criterion, showing how Pathwise methods following Young (1936) can be applied when 𝐻 > 1 2, while alternative approaches are needed for 𝐻 < 1 2.

• Chapter 4 extends beyond theoretical development to practical estimation methods, analysing both the rescaled range (R/S) approach introduced by Hurst (1951) and spectral regression methods following Geweke and Porter-Hudak(1983). We examine finite-sample biases, and asymptotic distributions of both methods providing rigorous theory for estimation.

Undergraduate_Mathematics_Dissertation_2025.pdf

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