Continuous-Time Red and Black: How to Control a Diffusion to a Goal

A player starts at x in (0, 1) and tries to reach 1. The process (X t , t ≥ 0) of his positions moves according to a diffusion process (or, more generally, an Ito process) whose infinitesimal parameters μ, σ are chosen by the player at each instant of time from a set depending on his current position. To maximize the probability of reaching 1, the player should choose the parameters so as to maximize μ/σ 2 , at least when the maximum is achieved by bounded, measurable functions. This implies that bold (timid) play is optimal for subfair (superfair), continuous-time red-and-black. Furthermore, in superfair red-and-black, the strategy which maximizes the drift coefficient of {log X t } minimizes the expected time to reach 1.