On the windings of complex-valued Ornstein-Uhlenbeck processes driven by a Brownian motion and by a Stable process.

Stavros Vakeroudis (1, 2)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires
2 School of Mathematics - Department of Mathematics [Manchester]

We deal with complex-valued Ornstein-Uhlenbeck (OU) process with parameter $\lambda\in\mathbb{R}$ starting from a point different from 0 and the way that it winds around the origin. The fact that the (well defined) continuous winding process of an OU process is the same as that of its driving planar Brownian motion under a new deterministic time scale (a result already obtained by Vakeroudis in \cite{Vak11}) is the starting point of this paper. We present the Stochastic Differential Equations (SDEs) for the radial and for the winding process. Moreover, we obtain the large time (analogue of Spitzer's Theorem for Brownian motion in the complex plane) and the small time asymptotics for the winding and for the process, and we deal with the exit time from a cone for a 2-dimensional OU process. Some Limit Theorems concerning the angle of the cone (when our process winds in a cone) and the parameter $\lambda$ are also presented. Furthermore, we discuss the decomposition of the winding process of complex-valued OU process in "small" and "big" windings, where, for the "big" windings, we use some results already obtained by Bertoin and Werner in \cite{BeW94}, and we show that only the "small" windings contribute in the large time limit. Finally, we study the windings of complex-valued OU process driven by a Stable process and we obtain the SDE satisfied by its (well defined) winding and radial process.

Mots clés: Complex-valued Ornstein-Uhlenbeck process; planar Brownian motion; windings; skew-product representation; exit time from a cone; Spitzer's Theorem; Stochastic Differential Equations; Bougerol's identity in law; Limit Theorems; radial and angular process; big and small windings; Lévy processes; Stable processes; isotropic Markov processes; subordination; Ornstein-Uhlenbeck processes driven by a Lévy process

Domaines: Probabilités [math.PR]

Stavros Vakeroudis. On the windings of complex-valued Ornstein-Uhlenbeck processes driven by a Brownian motion and by a Stable process.. 2012. ⟨hal-00733517v1⟩