Minimising the Time to Reach a Target and Return

Motivated in part by a problem in Metropolis-coupled, Markov chain Monte Carlo, we seek to minimise, in a suitable sense, the time it takes a (regular) diffusion with instantaneous reflection at 0 and 1 to travel from the origin to 1 and then return. The control mechanism is that we are allowed to chose the diffusion’s drift at each point in [0,1]. We consider the static and dynamic versions of this problem, where, in the dynamic version, we are only able to choose the drift at each point at the time of first visiting that point. The dynamic version leads to a novel type of stochastic control problem.