  We analyze the problem of the analytical characterization of the probability
distribution of financial returns in the exponential Ornstein-Uhlenbeck model
with stochastic volatility. In this model the prices are driven by a Geometric
Brownian motion, whose diffusion coefficient is expressed through an
exponential function of an hidden variable Y governed by a mean-reverting
process. We derive closed-form expressions for the probability distribution and
its characteristic function in two limit cases. In the first one the
fluctuations of Y are larger than the volatility normal level, while the second
one corresponds to the assumption of a small stationary value for the variance
of Y. Theoretical results are tested numerically by intensive use of Monte
Carlo simulations. The effectiveness of the analytical predictions is checked
via a careful analysis of the parameters involved in the numerical
implementation of the Euler-Maruyama scheme and is tested on a data set of
financial indexes. In particular, we discuss results for the German DAX30 and
Dow Jones Euro Stoxx 50, finding a good agreement between the empirical data
and the theoretical description.
