Stochastic Optimal Control for Hotel Dynamic Pricing: A Linear-Demand Formulation with Numerical Verification

We formulate hotel dynamic pricing as a finite-horizon stochastic optimal control problem with linear demand, recovering the original Gallego–van Ryzin (1994) framework with selfcontained derivations and numerical verification. The state is remaining room inventory; the control is the posted price; demand follows an inhomogeneous Poisson process with intensity λ(p, t) = μ(t)(a − bp)+. The optimal price admits the closed form p∗ = (¯p + ΔV )/2, where ¯p = a/b is the reservation price and ΔV is the shadow price of inventory — the arithmetic mean of the market ceiling and the inventory ceiling. We state structural properties of the value function (monotonicity, concavity, time-monotonicity), formally prove the saturation limit V → μ0aT ¯p/4 as C → ∞, and verify all claims numerically against direct Monte Carlo simulation (5/5 tests pass within 2σ). The two-regime structure (inventory-binding vs. demand-binding) is shown to transition near C ≈ μ0aT/2. We discuss a calibration procedure suited to small hotels with current ARI feeds and competitor-rate snapshots but no extensive booking history, with explicit assumptions stated