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Stochastic integration with respect to fractional Brownian motion and applications. (English) Zbl 1063.60080
González-Barrios, José M. (ed.) et al., Stochastic models. Seventh symposium on probability and stochastic processes, June 23–28, 2002, Mexico City, Mexico. Selected papers. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3466-5/pbk). Contemp. Math. 336, 3-39 (2003).
The article represents a good survey on several aspects of fractional Brownian motion, relevant stochastic analysis and applications. It consists of five sections. In the first section the author introduces fractional Brownian motion and its properties, in particular the non-semi-martingale property and integral representations. The next section treats stochastic calculus of variations with respect to fractional Brownian motion. Then the topic of stochastic integration with respect to fractional Brownian motion follows: first a review on different approaches appearing in the literature is given, then the relation of the divergence operator (defined in the previous section) and a path-wise stochastic integral is discussed. The latter is defined as the limit of integrals with respect to a regularization of fractional Brownian motion by convolution with a constant function. The cases of the Hurst index being above and below \(\frac{1}{2}\) are treated separately. Itô’s formulas in the two cases are also presented. In the fourth section stochastic differential equations driven by fractional Brownian motion are considered and existence and uniqueness of their solutions is established. The last section is devoted to applications, in particular vortex filaments and a fractional Black-Scholes model.
For the entire collection see [Zbl 1029.00031].

MSC:
60H05 Stochastic integrals
60H07 Stochastic calculus of variations and the Malliavin calculus
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60G18 Self-similar stochastic processes
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