Androshchuk T. Models in financial mathematics and mathematical statistics that involve fractional Brownian motion

The thesis is devoted to the study of three models from financial mathematics, mathematical statistics and econometrics, which are constructed with a help of fractional Brownian motion (fBm). For the mixed version of the Black-Scholes model of a fund market the absence of arbitrage in the class of self-financing strategies of a Markov type is proved. Also, the process of investor's capital under a self-financing strategy is presented as a limit of semimartingales. Dynamical system with small fractional Brownian noise is considered. Sufficient conditions under which the maximum likelihood estimator of unknown parameter is consistent and asymptotically normal are found. It is also proved that a family of probability measures generated by a system is locally asymptotically normal. For the unit-root bilinear model with fractional Gaussian noise the asymptotic behavior of the bilinear process is found under condition that the bilinear coefficient is asymptotically small. It is shown that the limit process is a solution of a certain stochastic differential equation with fBm. Auxiliary results are obtained about the construction of a sequence of absolutely continuous processes which converges in a certain sense to fBm. Also, convergence of the Euler approximation scheme with small deviations for a stochastic differential equation with fBm is proved.