Dynamic asset allocation and consumption ratcheting with costs

This paper delves into studying the optimal decisions regarding consumption and investment for an agent operating within a finite time frame, where any reduction in consumption is intolerable. In contrast to Jeon et al. (2018), we assume that the agent incurs a proportional utility cost with each increase in consumption. To tackle this problem, we employ the duality method to define the dual problem and investigate the resulting upper obstacle problem using PDE techniques. Our study reveals the existence of a critical time attributed to the utility cost, beyond which not only does the agent cease to further increase their consumption, but they also abstain from investing in the risky asset. Furthermore, the existence of the critical time makes free boundary analysis even more challenging, as this is because the presence of the critical time can also lead to the disruption of the monotonicity of the free boundary.