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Some aspects of stochastic calculus for the sub-fractional Brownian motion. (English) Zbl 1174.60024
The sub-fractional Brownian motion is a variant of the classical fractional Brownian motion. It is also a continuous centred Gaussian process, but with covariance function given by: ${s}^{2H}+{t}^{2H}-\frac{1}{2}\left[{\left(s+t\right)}^{2H}+{|s-t|}^{2H}\right]$, with Hurst parameter $0. It shares most properties of fractional Brownian motion, but does not have stationary increments. For example, it is not a semimartingale, and presents a long range dependence. And as the fractional Brownian motion too, it admits a pathwise adapted Wiener integral representation, by means of some kernel of fractional integral type. This allows the author to extend to this process the integral calculus of fractional Brownian motion, even in the anticipating case. Girsanov and Clark-Ocone formulas are then established. This study is finally applied to a detailed translation of Black-Scholes model and formula, the Brownian motion classically used being replaced by a sub-fractional Brownian motion.
##### MSC:
 60H05 Stochastic integrals