ABSTRACT
This chapter explains the structure of the internal history of a marked point process on R+. It discusses stochastic Stieltjes integrals. The chapter explains compensators of point processes and marked point processes and establish important properties. These include not only characterizations and explicit constructions but also applications to convergence in distribution and approximation of point processes. When the compensator is absolutely continuous, the derivative is a random process called the stochastic intensity. All integrals are simply stochastic Stieltjes integrals, because the integrators are either increasing processes or differences of increasing process. Specifically, the latter are usually innovation martingales. A central idea is the intimate relationship among martingales, predictable processes, and integrals with respect to martingales. In applications such as state estimation and martingale inference one must distinguish processes whose value for every t is determined by Ft-N from those for which error-free prediction of the current state from the strict past is impossible.
