A Finite Fuel Stochastic Control Problem on a Finite Time Horizon

Abstract

A player starts at x in $[0,a)$ with an initial amount of fuel $y > 0$ and seeks to reach the goal a by time $t_0 $ before spending all the fuel. Fuel is spent by the player at zero to keep the position nonnegative. The process $\{ X(t):0 \leq t \leq t_0 \} $ of the player’s position is an Ito process with reflection at zero, and its infinitesimal parameters $\mu $ and $\sigma $ are chosen by the player at each instant of time from a control set depending on the current position. The probability of reaching the goal a by the time $t_0 $ before exhausting all the fuel is maximized if the player can choose the parameters so that $\sigma $ and ${\mu / {\sigma ^2 }}$ are simultaneously maximized, at least when these maxima are sufficiently regular.As an application of this control problem, a new comparison theorem for Ito processes with reflection is derived.