Optimal dynamic risk sharing under the time‐consistent mean‐variance criterion

In this paper, we consider a dynamic Pareto optimal risk‐sharing problem under the time‐consistent mean‐variance criterion. A group of n insurers is assumed to share an exogenous risk whose dynamics is modeled by a Lévy process. By solving the extended Hamilton–Jacobi–Bellman equation using the Lagrange multiplier method, an explicit form of the time‐consistent equilibrium risk‐bearing strategy for each insurer is obtained. We show that equilibrium risk‐bearing strategies are mixtures of two common risk‐sharing arrangements, namely, the proportional and stop‐loss strategies. Their explicit forms allow us to thoroughly examine the analytic properties of the equilibrium risk‐bearing strategies. We later consider two extensions to the original model by introducing a set of financial investment opportunities and allowing for insurers' ambiguity towards the exogenous risk distribution. We again explicitly solve for the equilibrium risk‐bearing strategies and further examine the impact of the extension component (investment or ambiguity) on these strategies. Finally, we consider an application of our results in the classical risk‐sharing problem of a pure exchange economy.