Optimal investment and consumption under forward utilities with relative performance concerns

We study a n-player and mean-field portfolio optimization problem under relative performance concerns with non-zero volatility, for wealth and consumption. The consistency assumption defining forward relative performance processes leads to a sufficient characterization of such processes with mean of a Stochastic HJB equations, which highlights the link between wealth and consumption utility, and also characterizes the optimal strategies. In particular, forward relative performance processes with a wealth utility of CRRA type and separable time and space dependence necessarily have a consumption utility of the same form, with the same risk aversion parameter. This characterization gives a better understanding of the drift condition ensuring time consistency. In this setting, we establish closed form of the Nash equilibrium for both the n-player and mean eld problems. We also provide some numerical examples.