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

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.