Abstract
Consider a completely asymmetric Lévy process which has absolutely continuous transition probabilities. By harmonic transform, we establish the existence of the Lévy process conditioned to stay in a finite interval, called the confined process (the confined Brownian motion is F.B. Knight's Brownian taboo process). We show that the confined process is positive-recurrent and specify some useful identities concerning its excursion measure away from a point. We investigate the rate of convergence of the supremum process to the right-end point of the interval.
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