Stochastic differential reinsurance and investment games with delay under VaR constraints⋆

This article investigates stochastic differential reinsurance and investment games with delay under Value-at-Risk (VaR) constraints. In our model, three players and two games are considered, i.e., a non-zero-sum stochastic differential game between two insurers and a stochastic Stackelberg differential game between the reinsurer and the two insurers are investigated. The reinsurer can set the reinsurance premium price and invest her surplus in a financial market consisting of a risk-free asset and a risky asset. The two insurers can purchase proportional reinsurance and invest in the same financial market. Since the delay is considered in this article, the wealth processes of three players are described by the stochastic delay differential equations. In this article, we establish two optimization problems, one is to maximize the expected utility of the reinsurer's terminal wealth with delay, and the other is to maximize the expected utility of the combination of each insurer's terminal wealth and relative performance with delay. Furthermore, in order to control risk, we incorporate the VaR constraints into the optimizations and derive the corresponding value functions and Nash equilibrium strategies by using optimal control theory, dynamic programming principle, backward induction, Lagrange function and Karush-Kuhn-Tucker condition. Finally, several numerical examples are given.