Shevchenko G. Stochastic analysis for fractional and multifractional processes in models with long memory.

The thesis is devoted to fractional and multifractional processes, which are used to model phenomena with long memory property. The first line of research is the development of stochastic analysis for fractional Brownian motion, which consists in constructing numerical and statistical methods for stochastic differential equations driven by fractional Brownian motion, and establishing Malliavin regularity of solutions to such equations. The second line of research is the elaboration of the theory of mixed stochastic differential equations, which consists in obtaining conditions for the unique solvability, integrability, as well as in constructing numerical methods to solve such equations and establishing the Malliavin regularity of their solutions. The third line of research is the development of the theory of fractional and multifractional processes, which consists in defining new multifractional processes with desired modelling qualities, establishing their pathwise properties such as the existence and regularity of local times, constructing their smooth approximations. The last line of research is the application of stochastic analysis to financial mathematics problems such as hedging contingent claims, finding criteria for absence of arbitrage, optimal exercise of financial contracts.