An S-transform approach to integration with respect to a fractional Brownian motion

Abstract

We give an elementary definition of the (Wick--)Itô integral with respect to a fractional Brownian motion using the expectation, the ordinary Lebesgue integral and the classical (simple) Wiener integral. Then we provide new and simple proofs of some basic properties of this integral, including the so-called fractional Itô isometry. We calculate the expectation of the fractional Itô integral under change of measure and prove a Girsanov theorem for the fractional Itô integral (not only for fractional Brownian motion). We then derive an Itô formula for functionals of a fractional Wiener integral. Finally, we compare our approach with other approaches that yield essentially the same integral.