Martingales and Financial Mathematics in Discrete Time. Benoîte de Saporta

spaces, f : Ω ↦ E and g : E ↦ G are two (

, ε) and (ε,

)-measurable mappings, respectively, then for any B ∈

, g⁻¹(B) ∈ ε and consequently,

      Thus, the composition g ∘ f is indeed measurable on (Ω,

) in (G,

).

1.2. Probability elements

      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.

      1.2.1. Probabilities

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

      DEFINITION 1.7.– A probability or probability measure, or law of probability or distribution over a probability space (Ω,

) is a measure with a total mass equal to 1. In other words, a probability over (Ω,

) is a mapping ℙ :

→ ℝ such that

       – for any A ∈ , ℙ(A) ≥ 0,

       – ℝ(Ω) = 1,

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

      The triplet (Ω,

, ℙ) is then called a probability space.

      EXAMPLE 1.7.– Ω is endowed with the coarse σ-algebra

= {∅, Ω}. Thus, the single probability measure on (Ω,

) is given by:

      EXAMPLE 1.8.– Let Ω = [0, 1] and =

=

([0, 1]) be the Borel σ-algebra of [0, 1]. If λ denotes the Lebesgue measure, then the mapping:

      is a probability measure on (Ω,

).

      EXAMPLE 1.9.– Let Ω be non-empty set such that card(Ω) < ∞, where card(Ω) denotes the cardinal of Ω, that is, the number of elements in Ω. Consider the mapping ℙ from

(Ω) onto [0, 1] such that for every

      The mapping ℙ is then a probability on (Ω,

(Ω)), said to be the uniform probability on Ω.

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

      PROPOSITION 1.2.– Let (Ω,

, ℙ) be a probability space and (A_n)_n∈ℕ be a sequence of events in

.

       – 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.

      DEFINITION 1.8.– Let (Ω,

, ℙ) be a probability space.

       – 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

       – Two σ-algebras and are independent if for any A ∈ and B ∈ , A and B are independent.

       – A family of sub-σ-algebra i ⊂ , i ∈ I is mutually independent if any family of events (Ai ∈ i, i ∈ I) is mutually independent.

      EXAMPLE 1.10.– We roll a six-faced die and write

       – A1 the event “the number obtained is even”; and

       – A2 the event “the number obtained is a multiple of 3” .

      The universe of possible outcomes is Ω = {1, 2, 3, 4, 5, 6} which has a finite number of elements and as all its elements have the same chance of occurring, we can endow it with the uniform probability . Since

       we have

      Therefore, A₁ and A₂ are two independent events.

      EXAMPLE 1.11.– A coin is tossed twice. The following events are considered:

       – A1 “Obtaining tails (T) on the first toss”;

       – A2 “Obtaining heads (H) on the second toss”; and

       – A3 “Obtaining the same face on both tosses”.

       The universe of possible