In this paper, we investigate the pricing problem of a pure endowment contract when the insurance company has a limited information on the mortality intensity of the policyholder. The payoff of this kind of policies depends on the residual life time of the insured as well as the trend of a portfolio traded in the financial market, where investments in a riskless asset, a risky asset and a longevity bond are allowed. We propose a modeling framework that takes into account mutual dependence between the financial and the insurance markets via an observable stochastic process, which affects the risky asset and the mortality index dynamics. Since the market is incomplete due to the presence of basis risk, in alternative to arbitrage pricing we use expected utility maximization under exponential preferences as evaluation approach, which leads to the so-called indifference price. Under partial information this methodology requires filtering techniques that can reduce the original control problem to an equivalent problem in complete information. Using stochastic dynamics techniques, we characterize the indifference price of the insurance derivative in terms of the solutions of two backward stochastic differential equations. Finally, we discuss two special cases where we get a more explicit representation of the indifference price process.

中文翻译：

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部分信息下通过 BSDE 对纯禀赋的无差异定价

在本文中，我们研究了当保险公司对投保人的死亡强度信息有限时，纯养老合同的定价问题。此类保单的收益取决于被保险人的剩余寿命以及在金融市场上交易的投资组合的趋势，金融市场允许投资于无风险资产、风险资产和长寿债券。我们提出了一个建模框架，该框架通过可观察的随机过程考虑了金融和保险市场之间的相互依赖，这会影响风险资产和死亡率指数动态。由于存在基差风险，市场是不完整的，除了套利定价，我们使用指数偏好下的预期效用最大化作为评估方法，这导致了所谓的冷漠价格。在部分信息下，这种方法需要过滤技术，可以将原始控制问题简化为完整信息中的等效问题。使用随机动力学技术，我们根据两个后向随机微分方程的解来表征保险衍生品的无差异价格。最后，我们讨论了两种特殊情况，在这些情况下，我们可以更明确地表示无差异价格过程。我们用两个后向随机微分方程的解来刻画保险衍生品的无差异价格。最后，我们讨论了两种特殊情况，在这些情况下，我们可以更明确地表示无差异价格过程。我们用两个后向随机微分方程的解来刻画保险衍生品的无差异价格。最后，我们讨论了两种特殊情况，在这些情况下，我们可以更明确地表示无差异价格过程。