Pascal processes and their characterization

Let ([Pi]t) be a counting process on + with the property that for any t, T with 0[less-than-or-equals, slant]t[less-than-or-equals, slant]T the distribution of [Pi]T given the past t is Pascal (negative binomial) with one parameter being [Pi]t+1 and the probability parameter depending only on t and T. Does such a process exist? If so, how is it characterized? Finally, what is the most convenient way to model such a process? These questions are motivated by the distinguished role of the Pascal distribution in finding explicit solutions of optimal selection problems based on relative ranks. We answer them completely.