ABSTRACT
The most important keyword for this monograph is “martingale”. One of the principal roles of the martingale in statistics is to serve as an important tool for building semimartingale models in a variety of application areas. This chapter aims to introduce readers to the “faces” of two useful statistical models based on semimartingales, so that the real image of our research subject is nurtured more clearly in our minds.
Before proceeding to the above objectives, some of the reasons why the martingale is so useful will be explained from two different perspectives in Section 1.1, using minimal mathematical formulas.
The semimartingale may be interpreted as a “stochastic process version” of a statistical regression model. To illustrate the implications of this interpretation, Section 1.2, which is the main part of this chapter, provides an overview of statistical modeling with semimartingales by building two types of practical models. The first model is the diffusion process model, which is constructed in a fairly intuitive way in Subsection 1.2.1. On the other hand, it will be explained in Subsection 1.2.2 that Cox's regression model, which is widely used in survival analysis, is indeed one of the semimartingale models based on the Doob-Meyer decomposition.
