Matrix dufresne identities

Abstract

© The Author(s) 2015. Published by Oxford University Press. We prove a version of the classical Dufresne identity for matrix processes. More specifically, we show that the inverse Wishart laws on the space of positive definite r × r matrices can be realized by ∫∞0 MsMTs ds in which t →MT is a drifted Brownian motion on GLr(ℝ). This solves a problem in the study of spiked random matrix ensembles which served as the original motivation for this result. Various known extensions of the Dufresne identity (and their applications) are also shown to have analogs in this setting. In particular, we identify matrix-valued diffusions built from Mt which generalize in a natural way the scalar processes figuring into the geometric L'evy and Pitman theorems of Matsumoto and Yor.