Martingales and Financial Mathematics in Discrete Time. Benoîte de Saporta
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Название: Martingales and Financial Mathematics in Discrete Time

Автор: Benoîte de Saporta

Издательство: John Wiley & Sons Limited

Жанр: Математика

Серия:

isbn: 9781119885023

isbn:

СКАЧАТЬ always signifies mutual independence and not pairwise independence.

      1.2.2. Random variables

      Let us now recall the definition of a generic random variable, and then the specific case of discrete random variables.

      DEFINITION 1.9.– Let (Ω,

, ℙ) be a probabilizable space and (E, ε) be a measurable space. A random variable on the probability space (Ω,
, ℙ) taking values in the measurable space (E, ε), is any mapping X : Ω → E such that, for any B in ε, X−1(B) ∈
; in other words, X : Ω → E is a random variable if it is an (
, ε)-measurable mapping. We then write the event “X belongs to B” by

image

      In the specific case where E = ℝ and = ε =

(ℝ), the mapping X is called a real random variable. If E = ℝd with d ≥ 2, and ε =
(ℝd), the mapping X is said to be a real random vector.

      EXAMPLE 1.12.– Let us return to the experiment where a six-sided die is rolled, where the set of possible outcomes is Ω = {1, 2, 3, 4, 5, 6}, which is endowed with the uniform probability. Consider the following game:

       – if the result is even, you win 10 ;

       – if the result is odd, you win 20 .

       This game can be modeled using the random variable defined by:

image

      This mapping is a random variable, since for any B ∈

({10, 20}), we have

image

      and all these events are in

(Ω).

      DEFINITION 1.10.– The distribution of a random variable X defined on (Ω,

, ℙ) taking values in (E, ε) is the mapping ℙX : ε → [0, 1] such that, for any B ∈ ε,

image

      The distribution of X is a probability distribution on (E, ε); it is also called the image distribution of ℙ by X.

      DEFINITION 1.11.– A random real variable is discrete if X(Ω) is at most countable. In other words, if X(Ω) = xi, i ∈ I, where I ⊂ ℕ . In this case, the probability distribution of X is characterized by the family

image

      EXAMPLE 1.13.– Uniform distribution: Let

, ℙ) such that X(Ω) = {x1, ..., xN } and for any i ∈ {1, ..., N },

image

      It is then said that X follows a uniform distribution on {x1, ..., xN }.

      EXAMPLE 1.14.– The Bernoulli distribution: Let p ∈ [0, 1]. Let X be a random variable on (Ω,

, ℙ) such that X(Ω) = {0, 1} and

image

      It is then said that X follows a Bernoulli distribution with parameter p, and we write X ∼

(p).

      The Bernoulli distribution models random experiments with two possible outcomes: success, with probability p, and failure, with probability 1 – p. This is the case in the following game. A coin is tossed N times. This experiment is modeled by Ω = {T, H}N, endowed with the σ-algebra of its subsets and the uniform distribution. For 1 ≤ n ≤ N, the mappings Xn from Ω onto ℝ are considered, defined by

image

      the number of tails at the nth toss. Thus, Xn, 1 ≤ n ≤ N, are random real variables in the Bernoulli distribution with parameter 1/2 if the coin is balanced.

      EXAMPLE 1.15.– Binomial distribution: Let p ∈ [0, 1],

, ℙ) such that X(Ω) = {0, 1, ..., N } and for any k ∈ {0, 1, ..., N },

image

      It is then said that X follows a binomial distribution with parameters N and p, and we write X ∼

(N, p).

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