Optimal Control of Favorable Games with a Time Limit

This paper shows how to optimally control a stochastic process if one seeks to reach a certain value before a fixed time without first hitting zero. The process has a drift in the favorable direction. Discrete time random walks, as well as continuous time diffusion processes, are considered. The controls available are such that zero variance means zero drift, and more variance means more drift. To be more precise, in the continuous time case it holds that dXₜ = σ (Xₜ ,t)(μ ₜ dt + dBₜ ), where σ is the control variable. As corollaries of the results, some interesting inequalities for stochastic processes are obtained.