Compensators and Martingales

Abstracts

This chapter contains the basic theory of probability measures on the space W of counting process paths and the space M of discrete counting measures. Compensators and compensating measures are defined using hazard measures, and by exploring the structure of adapted and predictable processes on W and M (a structure that is discussed in detail), it is shown how the various forms of compensators characterize probabilities on canonical spaces. Also, these probabilities are described by the structure of basic martingales. Stochastic integrals (all elementary) are discussed and the martingale representation theorem is established. It is shown (It ô’s formula) how processes adapted to the filtration generated by an RCM may be decomposed into a predictable process and a local martingale. Finally, there is a discussion of compensators and compensating measures for counting processes and random counting measures defined on arbitrary filtered probability spaces.

Much of the material presented in this chapter is essential for what follows and complete proofs are given for the main results. Some of these proofs are quite long and technical and rely on techniques familiar only to readers well acquainted with measure theory and integration. It should be reasonably safe to omit reading the proofs of e.g., the following results: Theorem 4.1.1 (the proof of the last assertion), Proposition 4.2.1, Theorem 4.3.2, Proposition 4.3.5, Proposition 4.5.1 (although it is useful to understand why (4.65) implies that f is constant) and Theorem 4.6.1.