🤖 AI Summary
This paper addresses the optimal market-making problem for a limit-order book incorporating alpha signals, formulated as a Hamilton–Jacobi–Bellman quasi-variational inequality (HJBQVI) coupling stochastic and impulse controls. To overcome the time-step restrictions and poor stability of conventional explicit numerical schemes, we propose, for the first time, an unconditionally stable implicit time discretization combined with a policy iteration algorithm. We rigorously establish its convergence to the unique viscosity solution via monotonicity–stability–consistency analysis and the comparison principle. The resulting method significantly enhances the robustness and computational reliability of market-making strategies under high volatility, enabling dynamic spread management and inventory risk hedging. It provides a provably convergent, efficient, and numerically stable framework for solving coupled stochastic–impulse control problems in electronic market making.
📝 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.