A New Approach for the Continuous Time Kyle-Back Strategic Insider Equilibrium Problem

ArXiv ID: 2506.12281 “View on arXiv”

Authors: Bixing Qiao, Jianfeng Zhang

Abstract

This paper considers a continuous time Kyle-Back model which is a game problem between an insider and a market marker. The existing literature typically focuses on the existence of equilibrium by using the PDE approach, which requires certain Markovian structure and the equilibrium is in the bridge form. We shall provide a new approach which is used widely for stochastic controls and stochastic differential games. We characterize all equilibria through a coupled system of forward backward SDEs, where the forward one is the conditional law of the inside information and the backward one is the insider’s optimal value. In particular, when the time duration is small, we show that the FBSDE is wellposed and thus the game has a unique equilibrium. This is the first uniqueness result in the literature, without restricting the equilibria to certain special structure. Moreover, this unique equilibrium may not be Markovian, indicating that the PDE approach cannot work in this case. We next study the set value of the game, which roughly speaking is the set of insider’s values over all equilibria and thus is by nature unique. We show that, although the bridge type of equilibria in the literature does not satisfy the required integrability for our equilibria, its truncation serves as a desired approximate equilibrium and its value belongs to our set value. Finally, we characterize our set value through a level set of certain standard HJB equation.

Keywords: Kyle-Back model, stochastic differential games, forward-backward SDEs, insider trading, equilibrium theory, General Markets

Complexity vs Empirical Score

• Math Complexity: 9.5/10
• Empirical Rigor: 1.5/10
• Quadrant: Lab Rats
• Why: The paper uses advanced mathematics including forward-backward SDEs, HJB equations, and measure-theoretic probability, with no empirical data, backtests, or implementation details, focusing purely on theoretical equilibrium characterization.

  flowchart TD
    A["Research Goal:<br>New approach for continuous time Kyle-Back model"] --> B["Method: Stochastic Differential Games<br>via Forward-Backward SDEs"]
    B --> C["Data/Inputs:<br>Insider information flow & Market model"]
    C --> D["Computational Process:<br>Solve coupled FBSDE system"]
    D --> E{"Findings & Outcomes"}
    E --> F["Theorem 1:<br>Unique equilibrium for small time horizon"]
    E --> G["Theorem 2:<br>Set Value characterization via HJB level set"]
    E --> H["Theorem 3:<br>Bridge-type equilibrium serves as approximation<br>inside the Set Value"]