Stochastic analysis

Stochastic Processes
David Nualart nualart@mat.ub.es

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Stochastic Processes
Probability Spaces and Random Variables

In this section we recall the basic vocabulary and results of probability theory. A probability space associated with a random experiment is a triple (Ω, F, P ) where: (i) Ω is the set of all possible outcomes of the random experiment, and it is called thesample space. (ii) F is a family of subsets of Ω which has the structure of a σ-field: a) ∅ ∈ F b) If A ∈ F , then its complement Ac also belongs to F c) A1 , A2 , . . . ∈ F =⇒ ∪∞ Ai ∈ F i=1 (iii) P is a function which associates a number P (A) to each set A ∈ F with the following properties: a) 0 ≤ P (A) ≤ 1, b) P (Ω) = 1 c) For any sequence A1 , A2 , . . . of disjoints sets in F (that is, Ai ∩Aj = ∅if i = j), P (∪∞ Ai ) = ∞ P (Ai ) i=1 i=1 The elements of the σ-field F are called events and the mapping P is called a probability measure. In this way we have the following interpretation of this model: P (F )=“probability that the event F occurs” The set ∅ is called the empty event and it has probability zero. Indeed, the additivity property (iii,c) implies P (∅) + P (∅) + · · · = P (∅). The setΩ is also called the certain set and by property (iii,b) it has probability one. Usually, there will be other events A ⊂ Ω such that P (A) = 1. If a statement holds for all ω in a set A with P (A) = 1, then it is customary 2

to say that the statement is true almost surely, or that the statement holds for almost all ω ∈ Ω. The axioms a), b) and c) lead to the following basic rules of theprobability calculus: P (A ∪ B) = P (A) + P (B) if A ∩ B = ∅ P (Ac ) = 1 − P (A) A ⊂ B =⇒ P (A) ≤ P (B). Example 1 Consider the experiment of flipping a coin once. Ω = {H, T } (the possible outcomes are “Heads” and “Tails”) F = P(Ω) (F contains all subsets of Ω) 1 P ({H}) = P ({T }) = 2 Example 2 Consider an experiment that consists of counting the number of traffic accidents at a given intersectionduring a specified time interval. Ω = {0, 1, 2, 3, . . .} F = P(Ω) (F contains all subsets of Ω) λk P ({k}) = e−λ (Poisson probability with parameter λ > 0) k! Given an arbitrary family U of subsets of Ω, the smallest σ-field containing U is, by definition, σ(U) = ∩ {G, G is a σ-field, U ⊂ G} . The σ-field σ(U) is called the σ-field generated by U. For instance, the σ-field generated by the open subsets (orrectangles) of Rn is called the Borel σ-field of Rn and it will be denoted by BRn . Example 3 Consider a finite partition P = {A1 , . . . , An } of Ω. The σ-field generated by P is formed by the unions Ai1 ∪ · · · ∪ Aik where {i1 , . . . , ik } is an arbitrary subset of {1, . . . , n}. Thus, the σ-field σ(P) has 2n elements. 3

Example 4 We pick a real number at random in the interval [0, 2]. Ω =[0, 2], F is the Borel σ-field of [0, 2]. The probability of an interval [a, b] ⊂ [0, 2] is P ([a, b]) = b−a . 2

Example 5 Let an experiment consist of measuring the lifetime of an electric bulb. The sample space Ω is the set [0, ∞) of nonnegative real numbers. F is the Borel σ-field of [0, ∞). The probability that the lifetime is larger than a fixed value t ≥ 0 is P ([t, ∞)) = e−λt . A randomvariable is a mapping Ω −→ R ω → X(ω) which is F-measurable, that is, X −1 (B) ∈ F , for any Borel set B in R. The random variable X assigns a value X(ω) to each outcome ω in Ω. The measurability condition means that given two real numbers a ≤ b, the set of all outcomes ω for which a ≤ X(ω) ≤ b is an event. We will denote this event by {a ≤ X ≤ b} for short, instead of {ω ∈ Ω : a ≤ X(ω) ≤ b}. • Arandom variable defines a σ-field {X −1 (B), B ∈ BR } ⊂ F called the σ-field generated by X. • A random variable defines a probability measure on the Borel σ-field BR by PX = P ◦ X −1 , that is, PX (B) = P (X −1 (B)) = P ({ω : X(ω) ∈ B}). The probability measure PX is called the law or the distribution of X. We will say that a random variable X has a probability density fX if fX (x) is a nonnegative...