Mean-field equilibrium of heterogeneous agents under market impact
Abstract
Although market participants generally have access to a common information set, they make decisions based on forecasts formed over heterogeneous horizons. Because market impact depends on aggregate positions rather than trader identities, these decisions feed back into prices through their collective effect. We introduce a linear mean-field model of this interaction. The observed price is decomposed into a martingale component, a common predictable signal represented by a Volterra process, and the market impact generated by aggregate positions. Agents take positions according to conditional forecasts of future signal increments and a fraction of anticipated aggregate impact over their respective horizons. Within a Gaussian-Volterra framework, we characterize equilibrium through a linear fixed-point equation for aggregate positions and establish existence and uniqueness under explicit conditions. At equilibrium, we identify a balance condition that cancels the direct transmission of the common signal to the observed price. We then study the limit in which agents fully account for market impact. Along a suitably scaled family of equilibria satisfying explicit conditions, the contributions of the predictable signal and the market impact cancel in the limit, and the observed price converges to its martingale component. For fractional-type signals and Gamma-distributed horizons, we further derive local H\"older bounds and identify the horizon distributions for which the observed price has local regularity compatible with that of Brownian motion.