Martingales and Weak Convergence

Abstract

The basic questions dealt with in this book concern approximations of relatively complex processes by simpler and more tractable processes. These simpler processes will usually be diffusions, reflected diffusions, or reflected jump-diffusions. They might be controlled or not controlled. They are of interest to facilitate numerical or mathematical analysis and practical design, or to expose the more important structural properties and parametric dependencies. Some of the basic processes and methods will be introduced in this and in the next chapter. The martingale and Wiener processes are among the most fundamental in stochastic analysis, and the basic ideas are reviewed in Section 1. In the course of the analysis in the succeeding chapters it will be necessary to characterize the various processes that are obtained as limits of the approximation procedures. Of particular concern are methods for verifying that such a process is a Wiener process, and some standard and useful methods are described in Subsection 1.2 and in Section 8. Section 2 gives the basic definitions of the Itô stochastic integral with respect to a Wiener process, in preparation for the discussion of stochastic differential equations in the next chapter. Poisson and other jump processes will occur, for example, in the state-dependent models in Chapter 8, when the servers are subject to interruptions, and in “impulsive” models of arrival processes. The necessary background is dealt with in Section 3, as is the martingale decomposition of a Poisson-type jump process. In Section 4 we state some important results concerning the decomposition of the square of a martingale, and these will be employed in the proofs that certain weak-sense limits are Wiener processes or stochastic integrals.