Broker-Trader Partial Information Nash-Equilibria
Xuchen Wu, Sebastian Jaimungal
TL;DR
The paper addresses Nash equilibria between a broker and an informed trader under partial information in a multi-asset dealer market, where the broker observes prices in a lit market with transient impact and the trader has full market information. It develops a convex-analysis framework to prove existence and uniqueness of the equilibrium for short time horizons and derives first-order optimality conditions that yield explicit best-response mappings under two filtrations $\mathbb F$ and $\mathbb G$, leading to a coupled system of filtered forward–backward SDEs. The equilibrium is shown to correspond to a fixed point of a contraction on a Banach space, and the authors provide a polynomial perturbation expansion in the transient-impact strength $h$ with remainder terms that vanish at rate $o(h^M)$, together with closed-form expressions for the expansion coefficients. These results give a rigorous basis for analyzing broker–informed-trader interactions with partial information and offer a practical pathway to approximate equilibria through higher-order expansions, with potential extensions to all time horizons and more complex impact kernels. The work advances understanding of multi-asset trading with differing information sets and provides tools for numerical approximation of equilibria in stochastic market models.
Abstract
We study partial information Nash equilibrium between a broker and an informed trader. In this setting, the informed trader, who possesses knowledge of a trading signal, trades multiple assets with the broker in a dealer market. Simultaneously, the broker offloads these assets in a lit exchange where their actions impact the asset prices. The broker, however, only observes aggregate prices and cannot distinguish between underlying trends and volatility. Both the broker and the informed trader aim to maximize their penalized expected wealth. Using convex analysis, we characterize the Nash equilibrium and demonstrate its existence and uniqueness. Furthermore, we establish that this equilibrium corresponds to the solution of a nonstandard system of forward-backward stochastic differential equations (FBSDEs) that involves the two differing filtrations. For short enough time horizons, we prove that a unique solution of this system exists. Finally, under quite general assumptions, we show that the solution to the FBSDE system admits a polynomial approximation in the strength of the transient impact to arbitrary order, and prove that the error is controlled.