Affine credit risk models under incomplete information

We consider the problem of computing some basic quantities such
as defaultable bond prices and survival probabilities in a credit risk
model according to the intensity based approach. We let the default
intensities depend on an external factor process that we assume is
not observable. We use stochastic filtering to successively update
its distribution on the basis of the observed default history. On
one hand this allows us to capture aspects of default contagion
(information-induced contagion). On the other hand it allows us
to evaluate the above quantities also in our incomplete information
context. We consider in particular affine credit risk models and
show that in such models the nonlinear filter can be computed via
a recursive procedure. This then leads to an explicit expression
for the filter that depends on a finite number of sufficient statistics
of the observed interarrival times for the defaults provided one
chooses an initial distribution for the factor process that is of the
Gamma type.