Macroscopic Market Making Games via Multidimensional Decoupling Field

TL;DR

This paper extends the macroscopic market making framework to stochastic games, revealing dimension reduction and well-posedness results through a multidimensional decoupling approach and characteristic equations.

Contribution

It introduces a multidimensional decoupling field and characteristic equation to analyze stochastic market making games, providing new well-posedness results.

Findings

Dimension reduction in linear case equilibrium

Introduction of a multidimensional characteristic equation

New well-posedness results for stochastic games

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.