Читать книгу Martingales and Financial Mathematics in Discrete Time онлайн

13 страница из 24

 – If μ is a measure over a probabilizable space (Ω, ), then the triplet (Ω, , μ) is called a measured space.

We will now review the concept of a probability measure or probability distribution, and the concept of random variable, as well as the chief properties of these concepts.

A probability measure or probability distribution is a finite measure whose total mass is equal to 1.

 – for any A ∈ , ℙ(A) ≥ 0,

 – ℝ(Ω) = 1,

 – for any sequence of pairwise disjoint events in , denoted by (An)n∈ℕ, we have

We will only review those properties of a probability that will be useful for this book.

nn∈ℕ

 – If (An)n∈ℕ is increasing (for the inclusion), then,

 – If (An)n∈ℕ is decreasing (for the inclusion), then,

We will now review the concept of independent events and σ-algebras.

 – Two events, A and B, are independent if ℙ(A ∩ B) = ℙ(A) × ℙ(B).

 – A family of events (Ai ∈ i, i ∈ I) is said to be mutually independent if for any finite family J ⊂ I, we have