Krvavych Y. Stochastic integrals and stochastic differential equations with respect to fractional Brownian motion, and their applications in financial mathematics

The thesis is devoted to the developement of the main elements of analysis for stochastic integrals with integrator defined by the fractional Brownian motion (fBm), for fractional integrals with kernels defined by fBm. Especially in the thesis the stochastic Fubini theorems for fractional integrals with kernels defined by Wiener integrals with respect to (w.r.t.) fBm or defined by stochastic integrals with random integrand and fBm as integrator are proved, the differentiability conditions for those integrals and also the upper and lower maximal inequalities for moments of Wiener integrals w.r.t. fBm are found. All of these results were used further in the thesis under investigating of the existance and uniqueness conditions of global solution of the stochastic differential equations containing differential w.r.t. fBm and representing pure and mixed fractional stock price models; they were also used under investigating of the arbitrage possibilities, equilibrium conditions of the financial market, and u nder changing of the main probability measure.