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

We give an elementary definition of the (Wick-)Ito 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 Ito isometry. We calculate the expectation of the fractional Ito integral under change of measure and prove a Girsanov theorem for the fractional Ito integral (not only for fractional Brownian motion). We then derive an Ito formula for functionals of a fractional Wiener integral. Finally, we compare our approach with other approaches that yield essentially the same integral.