Optimal consumption and portfolio selection problem with downside consumption constraints

We study a general optimal consumption and portfolio selection problem of an infinitely-lived investor whose consumption rate process is subjected to downside constraint. That is, her consumption rate is greater than or equals to some positive constant. We obtain the general optimal policies in an explicit form using martingale method and Feynman–Kac formula. We derive some numerical results of optimal consumption and portfolio in the special case of a constant relative risk aversion (CRRA) utility function.