Article

# Bougerol's identity in law and extensions

• ##### Stavros Vakeroudis
Probability Surveys 01/2012; 9. DOI: 10.1214/12-PS195
Source: arXiv

ABSTRACT We present a list of equivalent expressions and extensions of Bougerol's
celebrated identity in law, obtained by several authors. We recall well-known
results and the latest progress of the research associated with this celebrated
identity in many directions, we give some new results and possible extensions
and we try to point out open questions.

0 Bookmarks
·
60 Views
• Source
##### Article: On the windings of complex-valued Ornstein-Uhlenbeck processes driven by a Brownian motion and by a Stable process
[Hide abstract]
ABSTRACT: 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.
09/2012;