  Financial time series exhibit two different type of non linear correlations:
(i) volatility autocorrelations that have a very long range memory, on the
order of years, and (ii) asymmetric return-volatility (or `leverage')
correlations that are much shorter ranged. Different stochastic volatility
models have been proposed in the past to account for both these correlations.
However, in these models, the decay of the correlations is exponential, with a
single time scale for both the volatility and the leverage correlations, at
variance with observations. We extend the linear Ornstein-Uhlenbeck stochastic
volatility model by assuming that the mean reverting level is itself random. We
find that the resulting three-dimensional diffusion process can account for
different correlation time scales. We show that the results are in good
agreement with a century of the Dow Jones index daily returns (1900-2000), with
the exception of crash days.
