Abstract We work with fractional Brownian motion with Hurst index H 1/2. We show that the pricing model based on geometric fractional Brownian motion behaves to certain extend as a process with bounded variation. This observation is based on a new change of variables formula for a convex...

The long-memory Gaussian processes presented as the integrals and are considered. The fractional Brownian motion is a particular case when [phi],[psi],h are the power functions. The integrals Vt are transformed into Gaussian martingales. The Girsanov theorem for Bt is stated and the Hellinger...

A real harmonizable multifractional stable process is defined, its Hölder continuity and localizability are proved. The existence of local time is shown and its regularity is established.

For a mixed stochastic differential equation driven by independent fractional Brownian motions and Wiener processes, the existence and integrability of the Malliavin derivative of the solution are established. It is also proved that the solution possesses exponential moments.

We consider the homogeneous stochastic differential equation with unknown parameter to be estimated. We prove that the standard maximum likelihood estimate is strongly consistent under very mild conditions. The conditions for strong consistency of the discretized estimator are established as well.