$N$-player and Mean-field Games in Itô-diffusion Markets with Competitive or Homophilous Interaction

Abstract

In Itô-diffusion environments, we introduce and analyze N -player and common-noise mean-field games in the context of optimal portfolio choice in a common market. The players invest in a finite horizon and also interact, driven either by competition or homophily. We study an incomplete market model in which the players have constant individual risk tolerance coefficients (CARA utilities). We also consider the general case of random individual risk tolerances and analyze the related games in a complete market setting. This randomness makes the problem substantially more complex as it leads to (N or a continuum of) auxiliary "individual" Itô-diffusion markets. For all cases, we derive explicit or closed-form solutions for the equilibrium stochastic processes, the optimal state processes, and the values of the games.

Mean Field Portfolio Games

“…In addition to [21], many player portfolio games with mean field interaction have been studied in [6,7,8,9,16,20], where in [6,7] Reis and Platonov studied many player games with forward utilities, in [20] Lacker and Soret extended the CRRA model in [21] to include consumption, by using PDE approaches. In [16] Hu and Zariphopoulou studied the many player games in Itô-diffusion environment.…”

“…In addition to [21], many player portfolio games with mean field interaction have been studied in [6,7,8,9,16,20], where in [6,7] Reis and Platonov studied many player games with forward utilities, in [20] Lacker and Soret extended the CRRA model in [21] to include consumption, by using PDE approaches. In [16] Hu and Zariphopoulou studied the many player games in Itô-diffusion environment. The same as [21], the market incompleteness of our paper is due to the individual noise driven the stock price while the market incompleteness of [8,9] is due to the trading constraint, which makes each player does not have access to the entire financial market, and the incompleteness in [16] is due to stock coefficients driven by a Brownian motion that is correlated with the one driving the stock price.…”

“…In [16] Hu and Zariphopoulou studied the many player games in Itô-diffusion environment. The same as [21], the market incompleteness of our paper is due to the individual noise driven the stock price while the market incompleteness of [8,9] is due to the trading constraint, which makes each player does not have access to the entire financial market, and the incompleteness in [16] is due to stock coefficients driven by a Brownian motion that is correlated with the one driving the stock price. The methodology of our paper shares more similarities with [8,9], where Espinosa and Touzi, Frei and Reis studied games without individual noise but with trading constraint, also by a BSDE approach.…”

Nash equilibria for relative investors via no-arbitrage arguments

“…Forward utilities are of CARA and CRRA type, respectively. Generalizations of the financial market can be found in [18,13,15]. The first paper [18] considers a stochastic volatility model (CIR) and CRRA utility for two players.…”

“…They consider a CARA utility and Nash equilibria are characterized as solutions of FBSDEs. Finally, [15] deal with a common Itô-diffusion market for all agents which may be incomplete in case of CARA utilities or complete in case of random risk tolerance coefficients. Further papers among others are [9] where the problem with partial information and heterogeneous priors is considered and [22] which discusses more economical questions like the structure of equilibria and the effect of additional agents.…”