An Introduction to (Stochastic) Calculus with Respect to Fractional Brownian Motion

This survey presents three approaches to (stochastic) integration with respect to fractional Brownian motion. The first, a completely deterministic one, is the Young integral and its extension given by rough path theory; the second one is the extended Stratonovich integral introduced by Russo and Vallois; the third one is the divergence operator. For each type of integral, a change of variable formula or Ito formula is proved. Some existence and uniqueness results for differential equations driven by fractional Brownian motion are available except for the divergence integral. As soon as possible, these integrals are compared. Key words: Gaussian processes, Fractional Brownian motion, Rough path, Stochastic calculus of variations