We give both necessary and sufficient conditions for a random variable to be represented as a pathwise stochastic integral with respect to fractional Brownian motion with an adapted integrand. We also show that any random variable is a value of such integral in an improper sense and that such...

For anticipative stochastic differential equations with Skorohod and white noise integrals the existence of approximate (in the sense that the difference between the left- and the right-hand sides of the equation is small in appropriate norm) solutions is proved under fairly mild conditions.

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 a stochastic differential equation involving a pathwise integral with respect to fractional Brownian motion. The estimates for the Hurst parameter are constructed according to first- and second-order quadratic variations of observed values of the solution. The rate of convergence of...

We study the possibility to control the moments of Wiener integrals of fractional Brownian motion with respect to the Lp- norm of the integrand. It turns out that when the self-similarity index , we can have only an upper inequality, and when we can have only a lower inequality.

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.

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...