ABSTRACT
The chapter starts with defining stochastic integrals and investigating their properties. We first give the definition of the stochastic integral of a predictable process with respect to a locally square-integrable martingale. A formula for the quadratic variation of stochastic integral is also provided. This formula will be frequently used as a fundamental tool in the latter part of this monograph to study the statistical analysis of stochastic processes.
In the middle part of this chapter, an intuitive explanation of Ito's formula will be given. A (special) semimartingale is a stochastic process that can be written in the form of the sum of a predictable process with finite-variation and a local martingale. The main reason why semimartingales are so important is that they are given in the additive of two good processes. On the other hand, some stochastic processes appearing in applications are not of an additive form, and the treatment for such processes may look difficult at first sight. Ito's formula is a powerful tool to transform a smooth functional of semimartingales into an additive form that is easy to analyze.
The chapter finishes with presenting Girsanov's theorem, which provides the likelihood ratio processes for semimartingales.
