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In this paper, we consider an optimal reinsurance problem. The surplus model of the insurance company is approximated by a diffusion model with the drift coefficient μ. The insurance company employs reinsurance to reduce the risk. π is the proportion of each claim paid by the company and the remainder proportion of the claim is paid by the reinsurer. λ(1 − π) is the rate at which the premiums are diverted to the reinsurer, thus it holds λ ≥ μ > 0 in general. We discuss two cases: (i) non-cheap reinsurance (when λ > μ), (ii) cheap reinsurance (when λ = μ). The objective of the insurance company is to make an optimal decision on reinsurance to reach a goal (a given surplus level) in minimal expected time. We disclose that for some case it is not suitable to search optimal decisions by minimizing the expected time to reach a goal. In order to deal with this case, we study two other methodologies (ruin probability minimization and expected discount factor maximization) for the optimal strategy selection problem in reinsurance decision.

We consider an insurance company whose surplus is represented by the classical Cramer-Lundberg process. The company can invest its surplus in a risk free asset and in a risky asset, governed by the Black-Scholes equation.There is a constraint that the insurance company can only invest in the risky asset at a limited leveraging level; more precisely, when purchasing, the ratio of the investment amount in the risky asset to the surplus level is no more than a; and when shortselling, the proportion of the proceeds from the short-selling to the surplus level is no more than b. The objective is to find an optimal investment policy that minimizes the probability of ruin. The minimal ruin probability as a function of the initial surplus is characterized by a classical solution to the corresponding Hamilton-Jacobi-Bellman (HJB) equation. We study the optimal control policy and its properties. The interrelation between the parameters of the model plays a crucial role in the qualitative behavior of the optimal policy. E.g., for some ratios between a and b, quite unusual and at first ostensibly counterintuitive policies may appear, like short-selling a stock with a higher rate of return to earn lower interest, or borrowing at a higher rate to invest in a stock with lower rate of return. This is in sharp contrast with the unrestricted case, first studied in Hipp and Plum (2000), or with the case of no shortselling and no borrowing studied in Azcue and Muler (2009).