The Skorokhod problem in a time-dependent interval

We consider the Skorokhod problem in a time-varying interval. We prove existence and uniqueness of the solution. We also express the solution in terms of an explicit formula. Moving boundaries may generate singularities when they touch. Under the assumption that the first time ττ when the moving boundaries touch after time zero is strictly positive, we derive two sets of conditions on the moving boundaries. We show that the variation of the local time of the associated reflected Brownian motion on [0,τ][0,τ] is finite under the first set of conditions and infinite under the second set of conditions. We also apply these results to study the semimartingale property of a class of two-dimensional reflected Brownian motions.