Stochastic Differential Games Between Two Insurers With Generalized Mean-Variance Premium Principle

Abstract

We study a stochastic differential game problem between two insurers, who invest in a financial market and adopt reinsurance to manage their claim risks. Supposing that their reinsurance premium rates are calculated according to the generalized mean-variance principle, we consider the competition between the two insurers as a non-zero sum stochastic differential game. Using dynamic programming technique, we derive a system of coupled Hamilton–Jacobi–Bellman equations and show the existence of equilibrium strategies. For an exponential utility maximizing game and a probability maximizing game, we obtain semi-explicit solutions for the equilibrium strategies and the equilibrium value functions, respectively. Finally, we provide some detailed comparative-static analyses on the equilibrium strategies and illustrate some economic insights.

Optimal reinsurance under the mean–variance premium principle to minimize the probability of ruin

“…The mean-variance premium principle combines the expected-value and variance premium principles; therefore, it is more general than either and includes each as a special case. Under the meanvariance premium principle, Zhang, Meng, and Zeng [30] studied optimal investment and reinsurance problems, Chen, Yang, and Zeng [9] studied a stochastic differential game between two insurers who invest in a financial market and adopt reinsurance to manage their claim risks, and Han, Liang, and Young [16] determined the optimal reinsurance strategy to minimize the probability of drawdown.…”

A non-zero-sum reinsurance-investment game with delay and asymmetric information

“…Under the objective of maximizing the utility of the relative performance, [6] studied a non-zero-sum stochastic differential investment and reinsurance game between two insurers whose surplus processes were modulated by continuous-time Markov chains; [37] considered a reinsurance and investment game between two insurers who have different opinions about some extra information. More studies on stochastic differential game problem may be found in [31], [33], [21], [14], [13], etc. In this paper, we consider a non-zero-sum stochastic differential reinsurance and investment game problem between two insurers.…”

“…Refer to [20,8,19,4,11,13] etc., the classic Cramér-Lundberg risk model can be approximated by the following diffusion process:…”

Optimal Strategies for an Ambiguity-Averse Insurer under a Jump-Diffusion Model and Defaultable Risk

“…At any time t, with a larger a, the insurer reduces expenses on reinsurance and pays a larger proportion of each claim by himself/ herself. Specially, when a � 1/2(1 + η)θ 1 , H(a, Z i ) � Z i , that is, the insurer pays all of the claims by himself/herself; when a � 0, he/she transfers all of the claims to the reinsurer according to Chen et al [41].…”