Some Problems in Stochastic Control Theory Related to Inventory Management and Coarsening.

Abstract

In this dissertation, we study two stochastic control problems arising from inventory management and coarsening.

First, we study a stochastic production/inventory system with a finite production capacity and random demand. The cumulative production and demand are modeled by a two-dimensional Brownian motion process. There is a setup cost for switching on the production and a convex holding/shortage cost, and our objective is to find the optimal production/inventory control that minimizes the average cost. Both lost-sales and backlogging cases are studied. For the lost-sales model we show that, within a large class of policies, the optimal production strategy is either to produce according to an $(s,S)$ policy, or to never turn on the machine at all (thus it is optimal for the firm to not do the business); while for the backlog model, we prove that the optimal production policy is always of the $(s, S)$ types. Our approach first develops a lower bound for the average cost among a large class of non-anticipating policies, and then shows that the value function of the desired policy reaches the lower bound. The results offer insights on the structure of the optimal control policies as well as the interplay between system parameters.

Then, we study a diffusive Carr-Penrose model which describes the phenomenon of coarsening. We show that the solution and the coarsening rate of the diffusive model converge to the classical Carr-Penrose model. Also, we demonstrate the relationship between the log concavity of the initial condition and the coarsening rate of the system. Under the assumption that the initial condition is log concave, there exists a constant upper bound on the coarsening rate of the diffusive problem. Our approach involves a representation of the solution using Dirichlet Green's function. To estimate this function, we exploit the property of a non-Markovian Gaussian process and derive bounds (both upper and lower) on the ratio between the Dirichlet and the full space Green's functions. The results shed light on the connection between the classical and diffusive Carr-Penrose models, and characterize the coarsening phenomenon under small noise perturbation.