Optimal Investment Under Relative Performance Concerns

We consider the problem of optimal investment when agents take into account their relative performance by comparison to their peers. Given N interacting agents, we consider the following optimization problem for agent i, 1≤i≤N: supπi∈AiEUixxx(1-λi)XTπi+λixxxXTπi-X-Ti,πxxxxxx,where Ui is the utility function of agent i, πi his portfolio, Xπi his wealth, X-i,π the average wealth of his peers, and λi is the parameter of relative interest for agent i. Together with some mild technical conditions, we assume that the portfolio of each agent i is restricted in some subset Ai. We show existence and uniqueness of a Nash equilibrium in the following situations: -unconstrained agents, -constrained agents with exponential utilities and Black-Scholes financial market. We also investigate the limit when the number of agents N goes to infinity. Finally, when the constraints sets are vector spaces, we study the impact of the λis on the risk of the market.