The Merton Problem with a Drawdown Constraint on Consumption

In this paper, we work in the framework of the Merton problem [17] but we impose a drawdown constraint on the consumption process. This means that consumption can never fall below a fixed proportion of the running maximum of past consumption. In terms of economic motivation, this constraint represents a type of habit formation where the investor is reluctant to let his standard of living fall too far from the maximum standard achieved to date. We state our candidate value function and optimal controls then provide a rigorous verification argument.