A BANG-BANG STRATEGY FOR A FINITE FUEL STOCHASTIC-CONTROL PROBLEM

The problem treated is that of controlling a process X(t) = x0 + integral-t/0-mu(s) ds + integral-t/0-sigma(s) dW(s) + A(t) with values in [0, al. The non-anticipative controls (mu(t), sigma(t)) are selected from a set C(x) whenever X(t-) = x and the non-decreasing process A(t) is chosen by the controller subject to the condition 0 less-than-or-equal-to A(t) less-than-or-equal-to y where y is a constant representing the initial amount of fuel. The object is to maximize the probability that X(t) reaches a. The optimal process is determined when the function rho(x) = sup {mu/sigma-2:(mu,sigma) is-an-element-of C(x)} has a unique minimum on [0, a] and satisfies certain regularity conditions. The optimal process is a combination of 'timid play' in which fuel is used gradually in the form of local time at 0, and 'bold play' in which all the fuel is used at once.