Abstract
We study a model of an insurance company whose surplus is represented by a pure diffusion. The company is allowed to buy proportional reinsurance and invest its surplus in a Black–Sholes financial market. Further, it is assumed that transaction cost rate of the reinsurance decreases linearly while the insurance company buys more reinsurance. We consider two optimization criteria – minimizing probability of ruin and maximizing expected exponential utility of terminal wealth for a fixed time. Corresponding Hamilton–Jacobi–Bellman (HJB) equations are analyzed; consequently we find explicit expressions of the minimal ruin probability, maximal expected terminal utility, and their associated optimal reinsurance–investment strategies via various parameter conditions. We observe from the explicit results that for some special parameter cases, the optimal investment–reinsurance strategies coincide under the two optimization criteria; i.e., Ferguson's longstanding conjecture on the relation between the two problems holds.