Fractional Brownian Motion and the Fractional Stochastic Calculus

TL;DR

This paper reviews the development of fractional Brownian motion and introduces a fractional stochastic calculus, including the definition of stochastic integrals with respect to fBm and their properties.

Contribution

It develops a formal framework for stochastic calculus with respect to fractional Brownian motion, extending classical Itô calculus to the fractional setting.

Findings

Defined and characterized stochastic integrals with respect to fBm.

Derived properties of fBm motivating the fractional stochastic calculus.

Presented a pathwise perspective for stochastic integration with fBm.

Abstract

This paper begins by giving an historical context to fractional Brownian Motion and its development. Section 2 then introduces the fractional calculus, from the Riemann-Liouville perspective. In Section 3, we introduce Brownian motion and its properties, which is the framework for deriving the It\^o integral. In Section 4 we finally introduce the It\^o calculus and discuss the derivation of the It\^o integral. Section 4.1 continues the discussion about the It\^o calculus by introducing the It\^o formula, which is the analogue to the chain rule in classical calculus. In Section 5 we present our formal definition of fBm and derive some of its properties that give motivation for the development of a stochastic calculus with respect to fBm. Finally, in Section 6 we define and characterize a stochastic integral with respect to fBm from a pathwise perspective.