We herein discuss the surplus process of an insurance company with various lines of business. The claim arrivals of the lines of business are modelled using multivariate point process with interdependencies between the marginal point processes, which depend only on the choice of thinning probabilities.
The insurer's aim is to maximize the expected exponential utility of terminal wealth by choosing an investment-reinsurance strategy, in which the insurer can continuously purchase proportional reinsurance and invest its surplus in a Black-Scholes financial market consisting of a risk-free asset and a risky asset.
We separately investigate the resulting stochastic control problem under unknown thinning probabilities, unknown claim arrival intensities and unknown claim size distribution for a univariate case.
We overcome the issue of uncertainty for these three partial information control problems using Bayesian approaches that result in reduced control problems, for which we characterize the value functions and optimal strategies with the help of the generalized Hamilton-Jacobi-Bellman equation, in which derivatives are replaced by Clarke's generalized gradients. ... mehrAs a result, we could verify that the proposed investment-reinsurance strategy is indeed optimal.
Moreover, we analysed the influence of unobservable parameters on optimal reinsurance strategies by deriving comparative results with the case of complete information, which shows a more risk-averse behaviour under more uncertainty. Finally, we provide numerical examples to illustrate the comparison results.