Application of the Pugachev–Sveshnikov equation to the Baxter occupation time problem

The Baxter problem, that is, an occupation (sojourn) time above a moving barrier, for a skew Brownian motion is considered. The latter is known as a model of a semipermeable barrier which permits either movement through it or reflection to the opposite direction with a specified probability. The Pugachev–Sveshnikov equation for a continuous Markov process is used to obtain an analytic solution of the problem. The generic method to solve the Pugachev–Sveshnikov equation for occupation-time type problems involves its reduction to a certain Riemann boundary value problem. This problem is solved, and the occupation time probability density function is derived. Along the way, some additional characteristics of the skew Brownian motion are obtained such as the first passage time, nonexceedance probability, occupation time moments, and some other characteristics.