Chapter 2

Probability, random variables and statistics

As seen, the modeling of randomness is one of the building blocks of mathematical finance. We recall here the basic notions on probability and random variables limiting the exposition to what will be really used in the rest of the book. The expert reader may skip this chapter and use it only as a reference and to align his own notation to the one used in this book.

Later on we will also discuss the problem of good calibration strategies of financial models, hence in this chapter we also recall some elementary concepts of statistics.

2.1 Probability

Random variables are functions of random elements with the property of being measurable. To make this subtle statement more precise we need the following preliminary definitions.

Definition 2.1.1

Let Ω be some set. A family of subsets of Ω is called σ-algebra on Ω if it satisfies the following properties:

i. ;

ii. if then its complementary set is in ;

iii. countable unions of elements ...