Implicit Numerical Scheme for the Hamilton-Jacobi-Bellman Quasi-Variational Inequality in the Optimal Market-Making Problem with Alpha Signal

Paper: arXiv 2512.20850

Authors: Alexey Meteykin

Abstract

We address the problem of combined stochastic and impulse control for a market maker operating in a limit order book. The problem is formulated as a Hamilton-Jacobi-Bellman quasi-variational inequality (HJBQVI). We propose an implicit time-discretization scheme coupled with a policy iteration algorithm. This approach removes time-step restrictions typical of explicit methods and ensures unconditional stability. Convergence to the unique viscosity solution is established by verifying monotonicity, stability, and consistency conditions and applying the comparison principle.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 2.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a rigorous mathematical framework for optimal market making with alpha signals using combined stochastic and impulse control (HJBQVI) and proves convergence for an implicit discretization policy iteration scheme. However, it is purely theoretical with zero empirical validation, real market data backtesting, or numerical experiments provided in the excerpt.

Research Flowchart

  flowchart TD
    A[Research Goal: Solve Market-Making HJBQVI<br>with Alpha Signal & Impulse Control] --> B[Model Setup: Limit Order Book Dynamics<br>& Stochastic/Impulse Control Formulation]
    B --> C[Implicit Time-Discretization Scheme<br>& Spatial Grid Discretization]
    C --> D[Policy Iteration Algorithm<br>for Coupled QVI System]
    D --> E[Convergence Proof:<br>Monotonicity, Stability & Consistency]
    E --> F[Outcomes: Unconditionally Stable Scheme<br>& Convergence to Unique Viscosity Solution]