A Bayesian approach for optimal reinsurance and investment in a diffusion model

A Bayesian adaptive control approach to the combined optimal investment/reinsurance problem of an insurance company is studied. The insurance company invests in a money market and a capital market index with an unknown appreciation rate, or “drift”. Using a Bayesian approach, the unknown drift is described by an unobservable random variable with a known (prior) probability distribution. We assume that the risk process of the company is governed by a diffusion approximation to the compound Poisson risk process. The company also purchases reinsurance. The combined optimal investment/reinsurance problem is formulated as a stochastic optimal control problem with partial observations. We employ filtering theory to transform the problem into one with complete observations. The control problem is then solved by the dynamic programming Hamilton–Jacobi–Bellman (HJB) approach. Semi-analytical solutions are obtained for the exponential utility case.