Market Making Spread Optimization

You are a market-maker in a continuous-time limit-order book for a single risky asset whose fundamental value v evolves as a Brownian motion with volatility σ. You must post a bid price B and an ask price A at every instant, creating a spread S = A − B. Your objective is to maximise the long-run expected profit rate subject to keeping inventory I(t) mean-reverting around zero. Each time a market order arrives it is filled at your quoted price; arrival rates are Poisson with intensity λ+(A − v) for buy orders and λ−(v − B) for sell orders, where λ± are decreasing functions of the distance to the fair value (adverse-selection cost). In addition you pay an inventory-penalty flow cost h(I) = ½γI² and a terminal quadratic penalty ½γI(T)² at horizon T. Derive the optimal feedback policy B*(t,I), A*(t,I) that maximises expected P&L net of penalties, and compute the resulting optimal spread S*(t,I). How does the spread depend on inventory, volatility, and the adverse-selection parameter κ = −λ′(0)? Provide closed-form or quasi-closed-form expressions in the infinite-horizon limit T → ∞.