  We investigate the optimal reinsurance problem in the risk model with jump
clustering features introduced in [7]. This modeling framework is inspired by
the concept initially proposed in [15], combining Hawkes and Cox processes with
shot noise intensity models. Specifically, these processes describe
self-exciting and externally excited jumps in the claim arrival intensity,
respectively. The insurer aims to maximize the expected exponential utility of
terminal wealth for general reinsurance contracts and reinsurance premiums. We
discuss two different methodologies: the classical stochastic control approach
based on the Hamilton-Jacobi-Bellman (HJB) equation and a backward stochastic
differential equation (BSDE) approach. In a Markovian setting, differently from
the classical HJB-approach, the BSDE method enables us to solve the problem
without imposing any requirements for regularity on the associated value
function. We provide a Verification Theorem in terms of a suitable BSDE driven
by a two-dimensional marked point process and we prove an existence result
relaying on the theory developed in [27] for stochastic Lipschitz generators.
After discussing the optimal strategy for general reinsurance contracts and
reinsurance premiums, we provide more explicit results in some relevant cases.
Finally, we provide comparison results that highlight the heightened risk
stemming from the self-exciting component in contrast to the externally-excited
counterpart and discuss the monotonicity property of the value function.
