  This paper deals with the filtering problem for a class of discrete time
stochastic volatility models in which the disturbances have rational
probability density functions. This includes the Cauchy distributions and
Student t-distributions with odd number of degrees of freedom. Using state
space realizations to represent the rational probability density functions we
are able to solve the filtering problem exactly. However the size of the
involved state space matrices grows exponentially with each time step of the
filter. Therefore we use stochastically balanced truncation techniques to
approximate the high order rational functions involved. In a simulation study
we show the applicability of this approach. In addition a simple method of
moments estimator is derived.
