Controlling a Process to a Goal in Finite Time

A player starts at x in (0, 1) and seeks to reach 1 by time t₀. The process {X(t), 0 ≤ t ≤ t₀} of the player's positions is a diffusion process (or an Ito process) whose infinitesimal parameters μ, σ are chosen by the player at each instant of time from a set depending on the current position. The probability of reaching 1 by time t₀ is maximized if the player can and does choose the parameters so that σ and μ/σ² are maximized, at least when these maxima are sufficiently regular. This result implies that bold play is optimal for subfair, continuous-time red-and-black and roulette when there is a limit on playing time.