Time consistent portfolio strategies for a general utility function
Oumar Mbodji
TL;DR
This work addresses time-inconsistent portfolio optimization under a general utility function in a complete market with non-constant discounting. It develops an extended HJB framework and introduces a marginal, utility-weighted discount rate $\rho(t,x)$ defined via a fixed-point problem, enabling time-consistent (subgame perfect) strategies to be characterized in feedback form. By transforming the marginal value function through a log-wealth change of variables, the authors reduce the problem to a linear parabolic PDE for the inverse function, and provide a concrete three-step algorithm to construct the subgame perfect strategies using Monte Carlo-inspired computation of $\rho$. The approach yields conditions under which the subgame perfect strategy coincides with the optimal strategy when discounting is replaced by $\rho$, and offers a practical computational pathway for general utilities in the finite-horizon Merton problem.
Abstract
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 $ρ(t,x)$ that depends on the time $t$ and wealth level $x$. A fixed point iteration is used to find $ρ$. The consumption to wealth ratio and the investment to wealth ratio are given in feedback form as functions of the value function.