Paper: arXiv 2406.05662

Abstract

Building on the macroscopic market making framework as a control problem, this paper investigates its extension to stochastic games. In the context of price competition, each agent is benchmarked against the best quote offered by the others. We begin with the linear case. While constructing the solution directly, the \textit{“ordering property”} and the dimension reduction in the equilibrium are revealed. For the non-linear case, we extend the decoupling approach by introducing a multidimensional \textit{“characteristic equation”} to analyse the well-posedness of the forward-backward stochastic differential equations. Properties of the coefficients in this characteristic equation are derived using tools from non-smooth analysis. Several new well-posedness results are presented.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 2.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper is dense with advanced mathematics, heavily utilizing forward-backward stochastic differential equations (FBSDEs), Riccati equations, and non-smooth analysis to establish global well-posedness of Nash equilibria. However, it lacks any mention of backtesting, statistical validation, or implementation details, focusing entirely on theoretical proof and model construction without empirical application.

Research Flowchart

  flowchart TD
  A["Research Goal: Model price competition<br>in macroscopic market making<br>as a stochastic game"] --> B["Methodology: Extend macroscopic<br>framework using decoupling field approach"]
  B --> C["Linear Case Analysis"]
  C --> D["Key Finding: Uncovered<br>Ordering Property &<br>Equilibrium Dimension Reduction"]
  B --> E["Non-Linear Case Analysis<br>with Multidimensional Characteristic Equation"]
  E --> F["Computational Process:<br>Non-smooth Analysis of Coefficients"]
  F --> G["Key Findings:<br>New Well-Posedness Results for FBSDEs<br>via Extended Decoupling Approach"]
  D --> G

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1 of 64 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.