🤖 AI Summary
This paper addresses the stochastic game modeling of macro-market makers under price competition, extending the classical macro-market-making framework to a multi-agent stochastic game where each market maker benchmarks its optimal quote against rivals’ best responses. Methodologically, it introduces a novel multidimensional decoupling field framework, integrating forward-backward stochastic differential equations (FBSDEs), nonsmooth analysis, and stochastic game theory to establish equilibrium ordering properties and intrinsic dimension reduction in the linear case; for the nonlinear case, it develops a multidimensional characteristic equation theory and rigorously proves existence, uniqueness, and well-posedness of solutions. Key contributions include: (i) the first scalable mathematical framework for macro-market-making games; (ii) identification and characterization of structural equilibrium properties—particularly quote ordering and dimension-reduction mechanisms; and (iii) several new well-posedness results, providing a rigorous theoretical foundation for high-dimensional market microstructure modeling.
📝 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 extit{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 extit{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.