Let X(t) = X(0) + B(t) + G U(t), t>0, where B is an n-dimensional Brownian motion, G a fixed matrix, and U a `singular control', i.e., an adapted process that is locally of bounded variation and keeps X in a given domain in R^n at all times t>0. Under some assumptions, a Hamilton-Jacobi-Bellman equation characterizes the value of a control problem in which a cost is to be minimized over all such controls U. The existence and uniqueness of solutions to this equation are shown to be connected to the intuitive `no arbitrage' condition, that states that an instantaneous control action that does not affect the process X does not result in any immediate reduction in cost.