Stochastic Integral and Covariation Representations for Rectangular Levy Process Ensembles

A Bercovici-Pata bijection Lambda(c) from the set of symmetric infinitely divisible distributions to the set of boxed plus(c)-free infinitely divisible distributions, for certain free convolution boxed plus(c) is introduced in Benaych-Georges (Random matrices, related convolutions. Probab Theory Relat Fields 144:471-515, 2009. Revised version of F. Benaych-Georges: Random matrices, related convolutions. arXiv, 2005). This bijection is explained in terms of complex rectangular matrix ensembles whose singular distributions are boxed plus(c)-free infinitely divisible. We investigate the rectangular matrix Levy processes with jumps of rank one associated to these rectangular matrix ensembles. First as general result, a sample path representation by covariation processes for rectangular matrix Levy processes of rank one jumps is obtained. Second, rectangular matrix ensembles for boxed plus(c)-free infinitely divisible distributions are built consisting of matrix stochastic integrals when the corresponding symmetric infinitely divisible distributions under Lambda(c) admit stochastic integral representations. These models are realizations of stochastic integrals of non-random functions with respect to rectangular matrix Levy processes. In particular, any boxed plus(c)-free selfdecomposable infinitely divisible distribution has a random matrix model of Ornstein-Uhlenbeck type integral(infinity)(0) e(-t)d Psi(t) where {Psi(t) : t >= 0} is a rectangular matrix Levy process.