Ralchenko K. Stochastic analysis and statistical estimation for fractional and related processes

The thesis is devoted to mathematical models with fractional Brownian motion and related processes. A multistable subordinator, a multifractional Poisson process, non-Euclidean generalizations of of the fractional Brownian field and the fractional Poisson field are introduced. Properties of these processes are investigated and limit theorems for them are obtained. We establish conditions for existence and uniqueness of a mild solution to stochastic heat equation with fractional noise and to a similar equation with white and fractional noises. The existence and uniqueness theorems for weak and strong solutions to a stochastic differential equation with stochastic volatility are proved, and strongly consistent drift parameter estimators for such equation are constructed. Also, we develop the methods of drift parameter estimation for a stochastic differential equation with fractional Brownian motion, the fractional and multifrctional Ornstein–Uhlenbeck process, and a Gaussian process with stationary increments.