  We consider an affine process $X$ which is only observed up to an additive
white noise, and we ask for its law, for some time $t > 0 $, conditional on all
observations up to this time $ t $. This is a general, possibly high
dimensional filtering problem which is not even locally approximately Gaussian,
whence essentially only particle filtering methods remain as solution
techniques. In this work we present an efficient numerical solution by
introducing an approximate filter for which conditional characteristic
functions can be calculated by solving a system of generalized Riccati
differential equations depending on the observation and the process
characteristics of the signal $X$. The quality of the approximation can be
controlled by easily observable quantities in terms of a macro location of the
signal in state space. Asymptotic techniques as well as maximization techniques
can be directly applied to the solutions of the Riccati equations leading to
novel very tractable filtering formulas. The efficiency of the method is
illustrated with numerical experiments for Cox-Ingersoll-Ross and Wishart
processes, for which Gaussian approximations usually fail.
