Exponential utility maximization for an insurer with time-inconsistent preferences

- A utility maximization problem is studied with a general discount function.
- Some of the parameters in the model are adapted stochastic processes.
- The methods of multi-person game and of martingale are used.
- A time-consistent equilibrium strategy is got by using BSDE and integral equation.

This paper studies the optimal consumption-investment-reinsurance problem for an insurer with a general discount function and exponential utility function in a non-Markovian model. The appreciation rate and volatility of the stock, the premium rate and volatility of the risk process of the insurer are assumed to be adapted stochastic processes, while the interest rate is assumed to be deterministic. The object is to maximize the utility of intertemporal consumption and terminal wealth. By the method of multi-person differential game, we show that the time-consistent equilibrium strategy and the corresponding equilibrium value function can be characterized by the unique solutions of a BSDE and an integral equation. Under appropriate conditions, we show that this integral equation admits a unique solution. Furthermore, we compare the time-consistent equilibrium strategies with the optimal strategy for exponential discount function, and with the strategies for naive insurers in two special cases.