In the paper we consider problem of estimating parameters entering drift a fractional Ornstein-Uhlenbeck type process in non-ergodic case, when underlying stochastic integral is ...

The present note deals with large time properties of the Lagrangian trajectories of a turbulent flow in IR and IR. We assume that the flow is driven by an incompressible time-dependent random velocity field with Gaussian statistics. We also assume that the field is homogeneous in space and stationary and Markovian in time. Such velocity fields can be viewed as (possibly infinite dimensional) Or...

The generalized Ornstein-Uhlenbeck process Vt = e −ξt ( V0 + ∫ t 0 edηs ) , t ≥ 0, driven by a bivariate Lévy process (ξt, ηt)t≥0 with starting random variable V0 independent of (ξ, η) fulfills the stochastic differential equation dVt = Vt−dUt + dLt for another bivariate Lévy process (Ut, Lt)t≥0, which is determined completely by (ξ, η). In particular it holds ξt = − log(E(U)t), t ≥ 0, where E(...

We consider the problem of option pricing under stochastic volatility models, focusing on the two processes known as exponential Ornstein-Uhlenbeck and Stein-Stein. We show they admit the same limit dynamics in the regime of low fluctuations of the volatility process, under which we derive the expressions of the characteristic function and the first four cumulants for the risk neutral probabili...

We consider Ornstein-Uhlenbeck processes (OU-processes) related to hypoelliptic diffusion on finite-dimensional Lie groups: let L be a hypoelliptic, left-invariant “sum of the squares”-operator on a Lie group G with associated Markov process X, then we construct OU-type processes by adding horizontal gradient drifts of functions U . In the natural case U(x) = − log p(1, x), where p(1, x) is the...

We study a least squares estimator b θT for the Ornstein-Uhlenbeck process, dXt = θXtdt+σdB H t , driven by fractional Brownian motion B H with Hurst parameter H ≥ 1 2 . We prove the strong consistence of b θT (the almost surely convergence of b θT to the true parameter θ). We also obtain the rate of this convergence when 1/2 ≤ H < 3/4, applying a central limit theorem for multiple Wiener integ...