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

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Highly relevant to quantitative trading, particularly for market-making strategies in high-frequency trading environments. The paper addresses the core problem of optimal quoting and inventory management with predictive signals (alpha). The implicit scheme's unconditional stability is practically important for real-time implementation. However, the paper is primarily theoretical/numerical and does not provide a ready-to-deploy trading system. The insights about optimal control behavior (quoting both sides when alpha is near zero, one-sided quoting when alpha is large, using market orders for extreme alpha) are directly applicable to market-making strategy design.

The numerical scheme requires careful implementation of: (1) implicit time discretization with upwind differencing for the drift term, (2) linear interpolation for shifted alpha values, (3) policy iteration with matrix assembly for each of the 4 limit order policies × 2 impulse directions × 2 impulse indicators, (4) backward time-stepping. The state space is (2Nα+1)×(2Nq+1) = 201×9 = 1809 grid points per time step. The policy iteration requires solving linear systems at each iteration. Expertise in numerical PDEs, stochastic control, and linear algebra is required. The mathematical framework (viscosity solutions, comparison principle) is advanced.

The paper provides explicit parameter values for the numerical experiment (T=10, A=300, Q=4, σ=0.01, θ=0.1, Δ=0.005, ε=0.005, λa=λb=1, k=200, ρ=1, γa=γb=60, ϕ=10^-6, ψ=0, Nα=101, N=200) and detailed algorithmic descriptions. However, no code repository or supplementary materials are provided. The mathematical formulations are complete and self-contained, enabling reimplementation by an expert in numerical PDEs.