Fractional Brownian Motion and the Fractional Stochastic Calculus

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 Itô integral. In Section 4 we finally introduce the Itô calculus and discuss the derivation of the Itô integral. Section 4.1 continues the discussion about the Itô calculus by introducing the Itô 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.