Multidimensional stochastic liquidity in Kyle's model of informed trading

2026-07-12

AI Summary

This work extends the Kyle model of informed trading to settings featuring stochastic liquidity and multiple assets, capturing the dynamic interplay between endogenous price impact and information revelation. By incorporating ideas from causal optimal transport, the authors formulate a variational framework that casts informed trading as an optimal liquidation problem driven by private information. Under martingale duality conditions, they establish a linear Gaussian equilibrium. The key theoretical innovation lies in introducing a matrix-valued martingale depth process, developing a Doob–Meyer decomposition for asymmetric matrix-valued submartingales, and reducing the general case to a coupled system of matrix-valued forward–backward stochastic differential equations (FBSDEs). While this FBSDE system poses the main analytical challenge, explicit solutions in the scalar and common eigenbasis cases confirm the validity and tractability of the proposed framework.

Abstract

We develop a variational formulation of Kyle's model of informed trading that accommodates stochastic liquidity and multiple traded assets. The main equilibrium result is stated first: under a martingale dual condition, a matrix-valued martingale depth process generates a linear-Gaussian equilibrium with stochastic matrix-valued price impact. We derive this martingale from a primal-dual problem, inspired by causal optimal transport, that characterizes the endogenous speed at which the insider injects private information into prices; in general, this problem admits only local martingale optimizers, and the martingale dual condition is the hypothesis that the optimizer is a true martingale. We interpret informed trading as the optimal liquidation of private information and verify the construction in the scalar and common-eigenbasis cases. The fully general matrix-valued case reduces to a coupled matrix FBSDE, which we isolate as the remaining obstruction. Along the way, we establish an independently interesting Doob-Meyer decomposition for general (not necessarily symmetric) matrix-valued submartingales.

Problem

• stochastic liquidity
• informed trading
• multiple assets
• Kyle's model
• price impact

Innovation

• stochastic liquidity
• matrix-valued martingale
• causal optimal transport
• linear-Gaussian equilibrium
• matrix FBSDE