In this paper we consider a portfolio optimization problem of the Merton's type over an infinite time horizon. Unlike the classical Markov model, we consider a system with delays. The problem is formulated as a stochastic control problem on an infinite time horizon and the state evolves according to a process governed by a stochastic process with delay. The goal is to choose investment and consumption controls such that the total expected discounted utility is maximized. Under certain conditions, we derive the explicit solutions for the associated Hamilton-Jacobi-Bellman (HJB) equations in a finite dimensional space for logarithmic and power utility functions. For those utility functions, verification results are established to ensure that the solutions are equal to the value functions, and the optimal controls are derived, too.

中文翻译：

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具有延迟的无限时间范围投资组合优化模型

在本文中，我们考虑了一个无限时间范围内的 Merton 类型的投资组合优化问题。与经典的马尔可夫模型不同，我们考虑具有延迟的系统。该问题被表述为无限时间范围内的随机控制问题，状态根据由具有延迟的随机过程控制的过程演变。目标是选择投资和消费控制，以使总预期贴现效用最大化。在某些条件下，我们为对数效用函数和幂效用函数在有限维空间中推导出关联的 Hamilton-Jacobi-Bellman (HJB) 方程的显式解。对于这些效用函数，建立验证结果以确保解与价值函数相等，并导出最优控制。