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. sup (pi i is an element of Ai) EUi((1 - lambda(i)) X-Tau(pi i) + lambda(i) (X-Tau(pi i) - (X) over bar (i,pi)(Tau))), where U-i is the utility function of agent i, pi(i) his portfolio, X-pi i his wealth, X-pi i, (X) over bar (i,pi) the average wealth of his peers, and lambda(i) 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 A(i). 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 lambda(i)s on the risk of the market.