An Envelope Theorem for Bayesian Dynamic Program and Its Application to an Inventory Problem

A generalized envelope theorem is established for a Bayesian dynamic program problem. An application of the theorem is given in a Bayesian inventory management problem with unobserved lost sales. Specifically, we show that the optimal inventory level with unobserved lost sales is greater than the optimal inventory level with observed lost sales. We prove this result under the continuous demand distribution, which complements the existing work in the literature. We further comment that the results can be easily extended to the Markov-modulated demand process.