Shalaiko T. Stochastic analysis of mixed models

The thesis is devoted to the investigation of models, that contain (multi)fractional Brownian motion and Wiener process. In particular it deals with the optimal mean square approximation of random variables with specific features of chaoses in the Ito-Wiener expansion, e.g. with non-degenerate first chaos, fractional Levy area. The strong localisability of a multifractional process that is sum of fractional and multifractional Browian motions (suitably scaled in time and space) is obtained. Next, we show that if a random variable is the final value of an adapted log-Holder continuous process, then it can be represented as a stochastic integral with respect to a fractional Brownian motion with an adapted integrand. Great attention is paid to the investigation of mixed stochastic differential equations, e.g. Malliavin calculus for such equations, applications of rough path analysis and investigation of the delayed version of the latter.