2 - Martingales, stopping times and random measures

Summary

Summary We begin by introducing the important concepts of filtration, martingale and stopping time. These are then applied to establish the strong Markov property for Lévy processes and to prove that every Lévy process has a càdlàg modification. We then meet random measures, particularly those of Poisson type, and the associated Poisson integrals, which track the jumps of a Lévy process. The most important result of this chapter is the Lévy–Itô decomposition of a Lévy process into a Brownian motion with drift (the continuous part), a Poisson integral (the large jumps) and a compensated Poisson integral (the small jumps). As a corollary, we complete the proof of the Lévy–Khintchine formula. We then obtain necessary and sufficient conditions for a Lévy process to be of finite variation and also to have finite moments. Finally, we establish the interlacing construction, whereby a Lévy process is obtained as the almost-sure limit of a sequence of Brownian motions with drift wherein random jump discontinuities are inserted at random times.

In this chapter, we will frequently encounter stochastic processes with càdlàg paths (i.e. paths that are continuous on the right and always have limits on the left). Readers requiring background knowledge in this area should consult Appendix 2.9 at the end of the chapter.