In this paper, we consider the utility maximization problem of an agent regarding optimal consumption-investment, job-switching strategy, and the optimal early retirement date. The agent can switch between two jobs or job categories at any time before retirement, but incurs a cost when switching to a position offering higher labor income. The agent's utility maximization involves a combination of stochastic control for consumption and investment, switching control for job-switching, and optimal stopping for early retirement decisions, making it a non-trivial and highly challenging problem. By utilizing the dynamic programming principle, we can derive the nonlinear Hamilton-Jacobi-Bellman (HJB) equation in the form of a system of variational inequalities with obstacle constraints, which arises from the agent's optimization problem. We employ guess and verify methods based on economic intuition to derive the closed-form solution of this HJB equation and demonstrate, through a verification theorem, that this solution aligns with the solution to the agent's utility maximization problem.

中文翻译：

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通过昂贵的工作转换选项做出最佳投资组合和退休决策


在本文中，我们考虑了代理在最佳消费投资、换工作策略和最佳提前退休日期方面的效用最大化问题。代理人可以在退休前的任何时间在两个工作或工作类别之间切换，但在切换到提供更高劳动收入的职位时会产生费用。代理的效用最大化涉及消费和投资的随机控制、换工作切换的切换控制以及提前退休决策的最佳停止的组合，使其成为一个非同小齐要且极具挑战性的问题。通过利用动态规划原理，我们可以推导出非线性 Hamilton-Jacobi-Bellman （HJB） 方程，其形式是具有障碍物约束的变分不等式系统，该方程由代理的优化问题产生。我们采用基于经济直觉的猜测和验证方法来推导出这个 HJB 方程的闭式解，并通过验证定理证明这个解与代理的效用最大化问题的解决方案一致。