Finite horizon portfolio selection problem with a drawdown constraint on consumption

In this paper we investigate the consumption and portfolio selection problem of a finitely-lived agent who faces drawdown constraint on consumption: the agent does not accept falling in her consumption below a fixed proportion of the historically highest level. We utilize the dual martingale method and study the dual problem, which can be transformed into an infinite series of optimal stopping problems. Based on a theory of partial differential equation (PDE), we analytically characterize a variational inequality arising from the optimal stopping problem. We provide a verification theorem that the value function for the original agent's problem is the Legendre-Fenchel transform of the integral of the value functions for the optimal stopping problems. Moreover, we derive integral equation representations for the optimal strategies and provide some numerical implications for the optimal strategies.