Martingales and Financial Mathematics in Discrete Time

1Elementary Probabilities and an Introduction to Stochastic Processes

This chapter reviews the basic concepts related to probability and random variables which will be useful for the rest of this text. For a more detailed explanation as well as demonstrations, the readers may refer to [BAR 07, DAC 82, FOA 03, OUV 08, OUV 09] in French and [BIL 12, CHU 01, DUR 10, KAL 02, SHI 00] in English. The readers who are already familiar with these concepts may proceed straight to section 1.3, which introduces the concept of stochastic processes.

This chapter begins with a brief summary of the concepts of a σ-algebra in section 1.1. These concepts will help in understanding the construction of the properties of conditional expectation in Chapter 2. We then study the chief definitions and properties of random variables and their distribution in section 1.2. There is an emphasis on discrete random variables as this entire book essentially studies discrete cases. Section 1.3 defines a stochastic process, which is the main subject studied in this book. Finally, there are exercises in handling these different concepts in section 1.4. The solutions are given in Chapter 8.

Throughout the rest of the text, Ω is a non-empty set and (Ω) denotes the set of the subsets of Ω :

The set Ω is called the ...