We introduce an extended version of the fractional Ornstein-Uhlenbeck (FOU) process where the integrand is replaced by the exponential of an independent Lévy process. We call the process the generalized fractional Ornstein-Uhlenbeck (GFOU) process. Alternatively, the process can be constructed from a generalized Ornstein-Uhlenbeck (GOU) process using an independent fractional Brownian motion (F...

First we give a construction of bridges derived from a general Markov process using only its transition densities. We give sufficient conditions for their existence and uniqueness (in law). Then we prove that the law of the radial part of the bridge with endpoints zero derived from a special multidimensional Ornstein-Uhlenbeck process equals the law of the bridge with endpoints zero derived fro...

The generalized Ornstein-Uhlenbeck process is derived from a bivariate Lévy process and is suggested as a continuous time version of a stochastic recurrence equation [1]. In this paper we consider the generalized Ornstein-Uhlenbeck process and provide sufficient conditions under which the process is exponentially ergodic and hence holds the exponentially β-mixing property. Our results can cover...

Bessel processes play an important role in financial mathematics because of their strong relation to financial processes like geometric Brownian motion or CIR processes. We are interested in the first time Bessel processes and more generally, radial Ornstein–Uhlenbeck processes hit a given barrier. We give explicit expressions of the Laplace transforms of first hitting times by (squared) radial...

The Gieseker-Uhlenbeck morphism maps the Gieseker moduli space of stable rank-2 sheaves on a smooth projective surface to the Uhlenbeck compactification, and is a generalization of the Hilbert-Chow morphism for Hilbert schemes of points. When the surface is the complex projective plane, we determine all the 1-point genus-0 Gromov-Witten invariants extremal with respect to the Gieseker-Uhlenbeck...

‎In this paper Lie symmetry analysis is applied in order to find new solutions for Fokker Plank equation of Ornstein-Uhlenbeck process‎. ‎This analysis classifies the solutions format of the Fokker Plank equation by using the Lie algebra of the symmetries of our considered stochastic process‎.

We study degenerate hypoelliptic Ornstein-Uhlenbeck operators in L spaces with respect to invariant measures. The purpose of this article is to show how recent results on general quadratic operators apply to the study of degenerate hypoelliptic Ornstein-Uhlenbeck operators. We first show that some known results about the spectral and subelliptic properties of Ornstein-Uhlenbeck operators may be...

We consider the α-stable Ornstein-Uhlenbeck process in Rd with the generator L = ∆α/2 − λx · ∇x. We show that if 2 > α ≥ 1 or α < 1 = d the Harnack inequality holds. For α < 1 < d we construct a counterexample that shows that the Harnack inequality doesn’t hold. keywords: α-stable Ornstein-Uhlenbeck process, harmonic functions, Harnack inequality. AMS Subject Classification: 60J75 60J45

This work is concerned with a quantization of the Pais-Uhlenbeck oscillators from the point of view of their multi-Hamiltonian structures. It is shown that the 2n-th order oscillator with a simple spectrum is equivalent to the usual anisotropic n dimensional oscillator. 1. This work is concerned with a quantization of the Pais-Uhlenbeck oscillators from the point of view of their multi-Hamilton...

Recently, there has been increasing interest in continuous-time stochastic models with jumps, a class of models which has applications in the fields of finance, insurance mathematics and storage theory, to name just a few. In this chapter we shall collect known results about a prominent class of these continuoustime models with jumps, namely the class of Lévy-driven Ornstein–Uhlenbeck processes...