  The Bercovici-Pata bijection maps the set of classical infinitely divisible
distributions to the set of free infinitely divisible distributions. The
purpose of this work is to study random matrix models for free infinitely
divisible distributions under this bijection. First, we find a specific form of
the polar decomposition for the L\'{e}vy measures of the random matrix models
considered in Benaych-Georges who introduced the models through their measures.
Second, random matrix models for free infinitely divisible distributions are
built consisting of infinitely divisible matrix stochastic integrals whenever
their corresponding classical infinitely divisible distributions admit
stochastic integral representations. These random matrix models are
realizations of random matrices given by stochastic integrals with respect to
matrix-valued L\'{e}vy processes. Examples of these random matrix models for
several classes of free infinitely divisible distributions are given. In
particular, it is shown that any free selfdecomposable infinitely divisible
distribution has a random matrix model of Ornstein-Uhlenbeck type
$\int_{0}^{\infty}e^{-t}d\Psi_{t}^{d}$, $d\geq1$, where $\Psi_{t}^{d}$ is a
$d\times d$ matrix-valued L\'{e}vy process satisfying an $I_{\log}$ condition.
