“…In particular, one needs to consider a family of optimal stopping problems of the form We shall show that the analogue of the relation (1.6), in the new context, is given by DQ(r, x, y) = u(r, x; DQ(., 0, y)) (1. 10) i.e., by a functional equation for the directional derivative in (1.9). We shall use refinements of the direct, probabilistic methodology of Karatzas and Shreve (1985), in order to establish (1.10); in particular, the device of 'switching paths at appropriate random times' will be used quite often, to compare expected costs at nearby points.…”