Mean-field equilibrium of heterogeneous agents under market impact

Paper: arXiv 2609.03115

Authors: Joseph Leclère, Mathieu Rosenbaum

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 Hölder bounds and identify the horizon distributions for which the observed price has local regularity compatible with that of Brownian motion.

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 1.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper presents a highly complex mathematical model for heterogeneous agents and market impact, demonstrating significant theoretical novelty. However, it lacks any empirical validation or data analysis, focusing purely on theoretical derivations and existence proofs. The clarity is good for a mathematically dense paper, but reproducibility is low due to the theoretical nature and absence of code or data.

Research Flowchart

  flowchart TD
    A[Research Goal: Model Heterogeneous Agents' Market Impact] --> B{Key Methodology: Linear Mean-Field Model};
    B --> C{Inputs: Common Info Set, Heterogeneous Horizons};
    C --> D[Process: Decompose Price, Agents' Position Decisions];
    D --> E{Computational: Gaussian-Volterra Framework, Fixed-Point Eq.};
    E --> F[Outcome 1: Equilibrium Characterization, Balance Condition];
    F --> G[Outcome 2: Limit Analysis - Price Converges to Martingale];
    G --> H[Outcome 3: Hölder Bounds & Price Regularity for Specific Signals/Horizons];

Cited References (15)

1. Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (1972)
2. Volterra Integral and Functional Equations (1990)
3. Stability of Operators and Operator Semigroups (2010)
4. Applied Probability and Queues (2003)
5. Necessary and Sufficient Conditions for H "o lder Continuity of G aussian Processes (2014)
6. Market impact as anticipation of the order flow imbalance (2015)
7. Approximation by mixtures of multivariate Erlang distributions (2026) — link
8. Th 'e orie de la sp 'e culation (1900)

0 of 15 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.