An introduction to (stochastic) calculus with respect to fractional Brownian motion. (English) Zbl 1126.60042

Coutin, Laure

Donati-Martin, Catherine (ed.) et al., Séminaire de Probabilités XL. Berlin: Springer (ISBN 978-3-540-71188-9/pbk). Lecture Notes in Mathematics 1899, 3-65 (2007).

Summary: 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 F. Russo and P. Vallois [Stochastics Stochastics Rep. 70, No. 1–2, 1–40 (2000; Zbl 0981.60053)]; the third one is the divergence operator. For each type of integral, a change of variable formula or Itô 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.
For the entire collection see [Zbl 1116.60002].

MSC:

60H05	Stochastic integrals
60G15	Gaussian processes

Keywords:

Gaussian processes; rough path; stochastic calculus of variations

Citations:

Zbl 0981.60053