Time-Consistent Mean-Variance Reinsurance-Investment Problems Under Unbounded Random Parameters: BSDE and Uniqueness
24 Pages Posted: 3 Jun 2018 Last revised: 29 Oct 2019
Date Written: January 23, 2019
Abstract
To strike the best balance between insurance risk and profit, insurers transfer insurable risk through reinsurance and enhance yield by participating into the financial market. The long-term commitment of insurance contracts makes insurers necessary to consider time-consistent (TC) reinsurance-investment policies. Using the open-loop TC mean-variance (MV) reinsurance-investment framework, we investigate the equilibrium reinsurance-investment problems for the financial market with unbounded random coefficients or, specifically, an unbounded risk premium. We characterize the problem via a backward stochastic differential equation (BSDE) framework. An explicit solution to the equilibrium strategies is derived for a constant risk aversion under a general class of stochastic models, embracing the constant elasticity of variance (CEV) and Ornstein-Uhlenbeck (OU) processes as special cases. For state-dependent risk aversions, the problem is related to the existence of a solution to a quadratic BSDE with unbounded parameters. A semi-closed form solution is derived, up to the solution to a nonlinear partial differential equation. By examining properties of the equilibrium strategies numerically, we find that the reinsurance decision is greatly affected by the market situation under the state-dependent risk aversion case. We prove the uniqueness of equilibrium strategies for both cases.
Keywords: Reinsurance-investment problem; time-inconsistency; backward stochastic differential equation (BSDE); mean-variance; Ornstein-Uhlenbeck (OU) process; constant elasticity of variance (CEV) process.
JEL Classification: G22, G11, C61
Suggested Citation: Suggested Citation