5. Subordinators at First Passage and Renewal Measures

In this chapter, we look at subordinators. These are Lévy processes which have paths that are non-decreasing. In addition, we consider killed subordinators, that is, subordinators which are sent to a “cemetery state” at an independent time that is exponentially distributed. Principally, we are interested in first passage over a fixed level and some asymptotic features thereof, as the level tends to infinity. In particular, we will study the (asymptotic) law of the overshoot and undershoot, as well as the phenomenon of crossing a level by hitting it. These three points of interest turn out to be very closely related to renewal measures. The results obtained in this chapter will be of significance later on when we consider first passage over a fixed level of a general Lévy process. As part of the presentation on asymptotic first passage, we will review some basic facts about regular variation. Regular variation will also be of use in later chapters. We conclude with a brief introduction to the theory of special subordinators which, amongst other things, permits the construction of a number of concrete examples of some of the theory discussed earlier in the chapter.