We consider an optimal investment, consumption, and life insurance purchase problem for a wage earner. We treat a stochastic factor model that the mean returns of risky assets depend linearly on underlying economic factors formulated as the solutions of linear stochastic differential equations. We discuss the partial information case that the wage earner can not observe the factor process and use only past information of risky assets. Then, our problem is formulated as a stochastic control problem with partial information. Applying the dynamic programming principle, we derive a coupled system of the Hamilton–Jacobi–Bellman (HJB) equation and two backward stochastic differential equations (BSDEs), and obtain the explicit solution. Finally, we strictly prove the verification theorem, and construct the optimal investment-consumption-insurance strategy.

中文翻译：

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具有部分信息的最优投资-消费-保险

我们考虑工薪阶层的最优投资、消费和人寿保险购买问题。我们处理一个随机因子模型，即风险资产的平均回报线性依赖于潜在的经济因素，这些因素被表述为线性随机微分方程的解。我们讨论工资收入者无法观察因子过程而仅使用风险资产的过去信息的部分信息情况。然后，我们的问题被表述为具有部分信息的随机控制问题。应用动态规划原理，我们推导了 Hamilton-Jacobi-Bellman (HJB) 方程和两个后向随机微分方程 (BSDE) 的耦合系统，并获得了显式解。最后，我们严格证明验证定理，