A Dynamic Programming Approach to Impulse Control of Brownian Motions

Abstract

This thesis considers an impulse control problem of a standard Brownian motion under a discounted criterion, in which every intervention incurs a strictly positive cost. The value function and an optimal $(\tau_{*}, Y_{*})$ policy are found using the dynamic programming principle together with the smooth pasting technique. The thesis also performs a sensitivity analysis by analyzing the limiting behaviors of the value function and the $(\tau_{*}, Y_{*})$ policy when the fixed intervention cost converges to zero. It is demonstrated that the limits agree with the classic fuel follower problem. The thesis next formulates and analyzes an $N$-player stochastic game of an impulse control problem under a discounted criterion. In the $N$-player stochastic game, each player controls an object. The objects are molded by an $N$-dimensional Brownian motion. A key aspect of the formulation is that each player aims to minimize her total impulse control cost and the total distance of her object to the moving center of the $N$ objects. The interaction mandates the players to closely follow each other's movements. The Nash equilibrium is characterized and analyzed by a system of Hamilton-Jacobi-Bellman equations. The case when $N=2$ is studied in detail.