Abstract:

This paper investigates the optimal reinsurance-investment strategies of two competing insurers under VaR constraints. We assume that the insurers' dynamic surplus processes are described by the classic Cramer-Lundberg (C-L) risk model, in which the premiums are determined by the loss-dependent premium principle. Moreover, the insurers can purchase proportional reinsurance and invest in a financial market consisting of a risk-free asset and a risky asset, where the price process of the risky asset is described by the geometric Brownian motion. Firstly, we aim to maximize the expected utility of the insurers' relative terminal wealth and then establish optimization problems with the VaR constraints. In the next, we solve the corresponding constrained optimization problems by using the optimal control theory and the dynamic programming principle. Specially, we get three different Nash equilibrium strategies under exponential utility. Finally, we illustrate the effects of some parameters on the optimal reinsurance strategy and the optimal investment strategy through specific numerical analysis, and find some interesting results.

Key words: VaR constraints, game, Lagrangian function, KKT condition, Nash equilibrium strategy