Some solvable stochastic control problemst^†

We find the explicit solution to several new problems in stochastic control, among them the finite-fuel problem of optimally tracking a standard Wiener process x+w t started at x by a nonanticipating process ξ t having ξ0=0 and total variation (fuel) so as to minimize the expected discounted cost . In n dimensions, the optimal process ξ is given thus: fuel is expended in a singular way to force x+w–ξ t reach and stay in the region remaining at time t, and f′ is a Bessel (n even) or an elementary function (n odd). Except for a possible initial jump in ξ the process is a degenerate diffusion that reflects at fixed angels off the boundary and is expressible in terms of the local times on the boundary components.