In this paper we propose a model for pricing GLWB variable annuities under a stochastic mortality framework. Our set-up is very general and only requires the Markovian property for the mortality intensity and the asset price processes. The contract value is defined through an optimization problem which is solved by using dynamic programming. We prove, by backward induction, the validity of the bang-bang condition for the set of discrete withdrawal strategies of the model. This result is particularly remarkable as in the insurance literature either the existence of optimal bang-bang controls is assumed or it requires suitable conditions. We assume constant interest rates, although our results still hold in the case of a Markovian interest rate process. We present extensive numerical examples, modelling the mortality intensity as a non mean reverting square root process and the asset price as an exponential Lévy process, and compare the results obtained for different parameters and policyholder behaviours.

在本文中，我们提出了一个在随机死亡率框架下为 GLWB 可变年金定价的模型。我们的设置非常通用，只需要死亡率强度和资产价格过程的马尔可夫性质。合同价值是通过使用动态规划解决的优化问题来定义的。我们通过逆向归纳法证明了模型的离散撤回策略集的爆炸条件的有效性。这一结果尤其引人注目，因为在保险文献中要么假设存在最优的爆炸式控制，要么需要适当的条件。我们假设利率不变，尽管我们的结果在马尔可夫利率过程的情况下仍然成立。我们提供了大量的数值例子，将死亡率强度建模为非均值回归平方根过程，将资产价格建模为指数利维过程，并比较不同参数和投保人行为获得的结果。