Time consistent portfolio strategies for a general utility function

We study the Merton portfolio management problem within a complete market, non constant time discount rate and general utility framework. The non constant discount rate introduces time inconsistency which can be solved by introducing sub game perfect strategies. Under some asymptotic assumptions on the utility function, we show that the subgame perfect strategy is the same as the optimal strategy, provided the discount rate is replaced by the utility weighted discount rate $\rho(t,x)$ that depends on the time $t$ and wealth level $x$. A fixed point iteration is used to find $\rho$. The consumption to wealth ratio and the investment to wealth ratio are given in feedback form as functions of the value function.