Two-limit Problems for Almost Semicontinuous Processes Defined on a Markov Chain

Problems related to the exit of a process with independent increments from an interval were investigated in many works (see, e.g. [1], [2]). Analogous problems were investigated for processes on a finite Markov chain under the semicontinuity condition [3, 4]. For walks on a countable Markov chain, a two-dimensional problem was studied in [5]. In the present paper, we consider distributions of some functionals associated with the exit from a bounded interval for a process with independent increments on a finite Markov chain under the assumption that this process crosses a positive level only by exponentially distributed jumps (an almost semicontinuous process [6]). The distributions of overjump functionals described by integral equations on a semi-axes are defined by the projection factorization method using an infinitely divisible factorization (instead of canonical factorization). In the present paper, we investigate functionals described by integral equations on an interval that can be extended to a semi-axes. In the solution of the extended equation, we use the method developed by Krein in [7, 8] and probability factorization identities. Consider a two-dimensional Markov process: