The Laguerre process and generalized Hartman–Watson law

Abstract

In this paper, we study complex Wishart processes or the so-called Laguerre processes $(X_t)_{t≥0}$. We are interested in the behaviour of the eigenvalue process; we derive some useful stochastic differential equations and compute both the infinitesimal generator and the semi-group. We also give absolute-continuity relations between different indices. Finally, we compute the density function of the so-called generalized Hartman–Watson law as well as the law of $T_0:=\inf {t, \det (X_t)=0}$ when the size of the matrix is $2$.