Optimal portfolio and retirement decisions with costly job switching options

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.