Mean field and N-agent games for optimal relative consumption-investment with jump risk and common noise

This paper studies an optimal consumption--investment problem for competitive agents in an \(N\)-player game and its associated mean field game.

Each agent invests in an individual risky asset subject to idiosyncratic noise, common noise and downward jump risk, and the interaction among agents is induced by relative performance concerns in both consumption and terminal wealth.

In the mean field limit, we characterize a deterministic mean field equilibrium in analytical form by using the stochastic maximum principle.

Numerical experiments are presented to illustrate the resulting equilibrium and its financial implications.

Finally, based on the obtained mean field equilibrium, we construct an approximate Nash equilibrium for the \(N\)-player game.

This model is motivated by \cite{Merton1971} and \cite{Lacker2020}.