  Let $X^{(\delta)}$ be a Wishart process of dimension $\delta$, with values in
the set of positive matrices of size $m$. We are interested in the large
deviations for a family of matrix-valued processes $\{\delta^{-1}
X_t^{(\delta)}, t \leq 1 \}$ as $\delta$ tends to infinity. The process
$X^{(\delta)}$ is a solution of a stochastic differential equation with a
degenerate diffusion coefficient. Our approach is based upon the introduction
of exponential martingales. We give some applications to large deviations for
functionals of the Wishart processes, for example the set of eigenvalues.
