Optimal Investment and Reinsurance for Insurers with Uncertain Time-Horizon

This paper considers the investment-reinsurance problems for an insurer with uncertain time-horizon in a jump-diffusion model and a diffusion-approximation model. In both models, the insurer is allowed to purchase proportional reinsurance and invest in a risky asset, whose expected return rate and volatility rate are both dependent on time and a market state. Meanwhile, the market state described by a stochastic differential equation will trigger the uncertain time-horizon. Specifically, a barrier is predefined and reinsurance and investment business would be stopped if the market state hits the barrier. The objective of the insurer is to maximize the expected discounted exponential utility of her terminal wealth. By dynamic programming approach and Feynman-Kac representation theorem, we derive the expressions for optimal value functions and optimal investment-reinsurance strategies in two special cases. Furthermore, an example is considered under the diffusion-approximation model, which shows some interesting results.