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 π i ∈ A iE U ix x x( 1 − λ i ) X T π i + λ ix x xX T π i − X ¯ T i , πx x xx x x, where U i is the utility function of agent i , π i his portfolio, X π ihis wealth,X ¯ i , πthe average wealth of his peers, and λ 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 λ i s on the risk of the market.