Introductory Stochastic Analysis for Finance and Insurance

CHAPTER 2

OVERVIEW OF PROBABILITY THEORY

In this chapter, we recall briefly some concepts and results in probability theory. We begin with probability spaces and information structures in Section 2.1. In some introductory probability textbooks, information structure and its association with a probability space is not discussed thoroughly. However, it is important to gain a good understanding of information structure in order to study stochastic calculus. Section 2.2 discusses random variables and their properties. Several common random variables and their distributions are given. Multivariate distributions are introduced in Section 2.3. Examples involving the multinomial distribution and bivariate normal distribution are given in this section. We consider conditional probability in Section 2.4 and conditional expectation in Section 2.5. Conditional probability and conditional expectation are extremely important since many results in stochastic calculus involve these notions. Section 2.6 concerns the Central Limit Theorem which is the foundation for the normal approximation.

Since this book focuses on stochastic processes, we do not discuss extensively the notions and results in probability theory. For a more thorough study of the topics covered in this chapter we refer readers to Hassett and Stewart (1999), Hogg and Craig (1978), Ross (1993), and Williams (1994). It is crucial to understand probability theory in order to master the materials in later chapters.

2.1   PROBABILITY ...