Portfolio and reinsurance optimization under unknown market price of risk

We investigate the optimal investment-and-reinsurance problem for insurance company with partial information on the market price of the risk. Through the use of filtering techniques, we convert the original optimization problem involving different filtrations into an equivalent stochastic control problem under the observation filtration, i.e. the so-called separated problem. The Markovian structure of the separated problem allows us to apply a classical approach to stochastic optimization based on the Hamilton–Jacobi–Bellman equation, and to provide explicit formulas for the value function and the optimal investment-and-reinsurance strategy. We finally discuss some comparisons between the optimal strategies pursued by a partially informed insurer and that followed by a fully informed insurer, and we evaluate the value of information using the idea of indifference pricing. These results are also supported by numerical experiments.