LEVEL-CROSSING TIMES FOR CERTAIN PROCESSES WITHOUT POSITIVE JUMPS

Let X be a process with cadlag trajectories, X-0 = 0, and let T-x (X) be its first crossing time of the level x. If x > 0 and X is a Levy pr ocess without positive jumps, A. A. BOROVKOV and V. M. ZOLOTAREV have expressed the law of T, (X) through the law of X. We show that the cla ss of processes satisfying a slightly less restrictive property is clo sed under weak convergence, and for some space and time transformation s. For a x less than or equal to 0 we give the law of T-x (X), X being the difference of a Levy process with increasing trajectories, and of a jump process of renewal type. This model is motivated by the ruin p roblem in risk theory.