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

TL;DR

This paper introduces an implicit numerical scheme for solving the Hamilton-Jacobi-Bellman quasi-variational inequality in a market-making context, enabling stable and efficient computation of optimal strategies.

Contribution

The paper develops a novel implicit time-discretization method combined with policy iteration for HJBQVI, ensuring unconditional stability and convergence in market-making models.

Findings

The scheme is unconditionally stable and converges to the viscosity solution.

It effectively handles the combined stochastic and impulse control problem.

The method improves computational robustness over explicit schemes.

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.