Time of exit of a semicontinuous process with boundary (Q1097594)

Let \((\xi (t))_{t\geq 0}\) be a homogeneous process with independent increments and positive jumps, the paths of which are right continuous. Loosely speaking, it is possible to define on the phase space [0,\(\infty)\) a homogeneous Markov process \((x(t))_{t\geq 0}\) as follows: Up to the moment the \(\xi\)-process reaches the null level, the increments of both processes coincide. Then the x-process remains in this state a positive time \(\tau_{0+}\) with a parameter 1/\(\mu\), and at a (random) moment \(\tau_{0+}\) leaves it, having a positive increment \(\xi\) (distributed according to a given function F) which does not depend on the previous history of the process. The aim of the paper is to study the limit of the repartition of the random variable \(\tau_ T=\inf \{t:x(t)\geq T\}\) as \(t\to \infty\).