On proportional reinsurance with a linear transaction rate

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. Corre- sponding 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 param- eter 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.