Article

# Bougerol's identity in law and extensions

01/2012; 9(1). 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.

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Available from: Stavros Vakeroudis, Oct 06, 2014
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• "For another two-dimensional extension of (1.1), and even a three-dimensional one we refer to Vakeroudis [13, Sections 4.2 and 4.3]. We are only interested in the following particular case of the identity (1.1) presented in [13] [14]. Bougerol's identity (1.1) is equivalent to the equality of the corresponding "
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ABSTRACT: We point out an easy link between two striking identities on exponential functionals of the Wiener process and the Wiener bridge originated by Bougerol, and Donati-Martin, Matsumoto and Yor, respectively. The link is established using a continuous one-parameter family of Gaussian processes known as $\alpha$-Wiener bridges or scaled Wiener bridges, which in case $\alpha=0$ coincides with a Wiener process and for $\alpha=1$ is a version of the Wiener bridge.
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##### Article: Some two-dimensional extensions of Bougerol's identity in law for the exponential functional of linear Brownian motion
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ABSTRACT: The main result is a two-dimensional identity in law. Let (B t ,L t ) and (β t ,λ t ) be two independent pairs of a linear Brownian motion with its local time at 0. Let A t =∫ 0 t exp(2B s )ds. Then, for fixed t, the pair (sinh(B t ),sinh(L t )) has the same law as (β(A t ),exp(-B t )λ(A t )), and also as (exp(-B t )β(A t ),λ(A t )). This result is an extension of an identity in distribution due to Bougerol that concerned the first components of each pair. Some other related identities are also considered.
Preview · Article · Jan 2012 · Revista Matematica Iberoamericana
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##### Article: On the windings of complex-valued Ornstein-Uhlenbeck processes driven by a Brownian motion and by a Stable process
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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.
Full-text · Article · Sep 2012 · Stochastics An International Journal of Probability and Stochastic Processes