We study an optimal reinsurance problem under the criterion of maximizing the expected utility of terminal wealth when the loss process exhibits jump clustering features and the insurance company has restricted information about the claims arrival intensity. By solving the associated filtering problem we reduce the original problem to a stochastic control problem under full information. Since the classical Hamilton-Jacobi-Bellman approach does not apply, due to the infinite dimensionality of the filter, we choose an alternative approach based on Backward Stochastic Differential Equations (BSDEs). Precisely, we characterize the value process and the optimal reinsurance strategy in terms of a BSDE driven by a marked point process. The talk is based on a joint work with M. Brachetta, G. Callegaro and C. Sgarra (arXiv:2207.05489, 2022).[-]