Practical Financial Optimization: Decision Making for Financial Engineers

Appendix B

Basics of Probability Theory

We give here some basic definitions on probability spaces. Additional background material on probability theory can be found, e.g., in Billingsley (1995) and – as applied to stochastic programming where the basics of probability theory are mainly needed in this book – in Kall (1976), and Wets (1989). Ãdenotes random variables which belong to some probability space as defined below.

B.1 Probability Spaces

Let Ξ be an arbitrary space or set of points. A σ-field for Ξ is a family ∑ of subsets of Ξ such that Ξ itself, the complement with respect to Ξ of any set in ∑, and any union of countably many sets in ∑ are all in ∑. The members of ∑ are called measurable sets, or events in the language of probability theory. The set Ξ with the σ-field ∑ is called a measurable space and is denoted by (Ξ,∑).

Let Ξ be a (linear) vector space and ∑ a σ-field. A probability measure P on (Ξ,∑) is a real-valued function defined over the family ∑, which satisfies the following conditions: (i) 0 ≤ P(A) ≤ 1 for A ∈ ∑ (ii) P(ø) = 0 and P(Ξ) = 1; and (iii) if is a sequence of disjoint sets A_l ∈ and if then . The triplet (Ξ, ∑, P) is called a probability space. The support ...