Law of two-sided exit by a spectrally positive strictly stable process

For a spectrally positive strictly stable process with index in (1, 2), we obtain (i) the sub-probability density of its first exit time from an interval by hitting the interval’s lower end before jumping over its upper end, and (ii) the joint distribution of the time, undershoot, and jump of the process when it makes the first exit the other way around. The density of the exit time is expressed in terms of the roots of a Mittag-Leffler function. Some theoretical applications of the results are given.