Some Aspect Of Fractional Ornstein-Uhlenbeck Process

From both theoretical and applied perspectives, fractional Brownian motion is challenging and of great interest. In fact, the fractional Brownian motion has already been successfully applied to hydrology, network traffic analysis, finance as well as various other fields. In this thesis, we consider the fractional Ornstein-Uhlenbeck process V_t~H (see [4, 10]), which is the solution of the Langevin equation driven by B~H.In the first one, we establish IP estimate for fractional Ornstein-Uhlenbeck process, and we show thatare equivalent for all stopping times τ of B~H and 0 < p < ∞,Furthermore, we point out that the fractional O-U process V~H has not the same finite dimensional distribution as Lamperti transform of B~H unless H =1/2 . As a related result, we also give the L~P estimate for the ratioIn the second setting, we study the two types of the local times of the fractional Ornstein-Uhlenbeck process with Hurst index 1/2 < H < 1, that is, local time andweighted local time. We give Tanaka formula for the process and some properties of local times.