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 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.