  We investigate the optimal reinsurance problem when the loss process exhibits
jump clustering features and the insurance company has restricted information
about the loss process. We maximize expected exponential utility of terminal
wealth and show that an optimal solution exists. By exploiting both the
Kushner-Stratonovich and Zakai approaches, we provide the equation governing
the dynamics of the (infinite-dimensional) filter and characterize the solution
of the stochastic optimization problem in terms of a BSDE, for which we prove
existence and uniqueness of solution. After discussing the optimal strategy for
a general reinsurance premium, we provide more explicit results in some
relevant cases.
