On proportional reinsurance with a linear transaction rate

Issue title: The Interplay between Finance and Actuarial Science

Article type: Research Article

Authors: Luo, Shangzhen

Affiliations: Department of Mathematics, University of Northern Iowa, Cedar Falls, IA 50614, USA

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.

Keywords: Ruin probability, exponential utility, HJB equation, proportional reinsurance, linear transaction rate

DOI: 10.3233/RDA-2011-0034

Journal: Risk and Decision Analysis, vol. 3, no. 1-2, pp. 115-137, 2012