Before revision:  Analogously to the case of ordinary optimal stopping problems for one-dimensional Brow- nian motions we reduce the problem to the geometric problem of finding the smallest majorant of the reward function in an two-parameter function space.
After revision:  Analogously to the case of ordinary optimal stopping problems for one-dimensional Brownian motions we reduce the problem to the geometric problem of finding the smallest majorant of the reward function in an two-parameter function space.