Optimal reinsurance in a diffusion setting

The business of insurance companies is to take on the risk of policyholders (individuals or companies), receiving in return the payment of a premium. In order to protect themselves against big losses, and not be at risk of insolvency, insurance companies usually reinsure part of their portfolio by transferring part of the risk taken to another insurance company. Reinsurance works, in this way, as the insurance of the insurer itself. The optimal reinsurance problem aims at answering two fundamental questions: (i) What type of reinsurance contract should be done; (ii) how much risk should be transferred to the reinsurance company. This master's final work, seeks to find the optimal reinsurance for 3 different optimality criterion: (i) minimizing the probability of ruin occurring in infinite time; (ii) maximizing the expected value; and (iii) minimizing the variance of the process. To obtain the optimal reinsurance treaty, the classic risk model of Crámer and Lundberg is approximated by a diffusion process which is described by a Brownian motion process. The surplus Brownian motion process is defined by parameters that incorporate several characteristics of the underlying Crámer-Lundberg process, including the different premium calculation principles and the different types of reinsurance treaties. In this work, the reinsurance treaties under study are the proportional quota-share treaty and the non-proportional excess of loss treaty, and the premium calculation principle considered by both, the first insurer and the reinsurer, is the expected value principle. After building the model, the probability of ruin is analysed. The present study addresses this moment of ruin, i.e., when the surplus process hits zero or negative values, in continuous time and infinite time horizon. The optimal reinsurance strategy is obtained numerically and a sensitivity analysis is made, using Mathematica software.