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OPTIMAL INVESTMENT UNDER RELATIVE PERFORMANCE CONCERNS
Author(s) -
Espinosa GillesEdouard,
Touzi Nizar
Publication year - 2015
Publication title -
mathematical finance
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.98
H-Index - 81
eISSN - 1467-9965
pISSN - 0960-1627
DOI - 10.1111/mafi.12034
Subject(s) - uniqueness , portfolio , mathematical economics , infinity , financial market , economics , investment strategy , limit (mathematics) , mathematical optimization , investment (military) , market portfolio , mathematics , exponential utility , econometrics , microeconomics , financial economics , finance , profit (economics) , mathematical analysis , politics , political science , law
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.