Student Seminar Talk PROBABILITY AND MARTINGALES

I will introduce some important concepts in probability theory: sub sigma-algebras, conditioning and martingale processes. We will focus on their intuitive meaning with a lot of examples. 1. Basic concepts A probability space is a triple (Ω,F , P ). The elements of F are called events. A random variable is a function X : Ω→ R which is measurable in the sense that X−1A = {X ∈ A} = {ω ∈ Ω : X(ω) ∈ A} ∈ F for every Borel set A ⊂ R. In short notation, X−1B(R) ⊂ F where B(R) is the Borel σ-algebra of R. The σ-algebra generated by X is σ(X) = X−1B(R), this is the smallest sub σ-algebra of F with respect to which X is measurable. 2. Sub σ-algebras 2.1. Intuitive meaning. A sub σ-algebra G ⊂ F represents partial information, in the following sense. We think that ω ∈ Ω has been chosen randomly (according to the probability measure P ), and someone now tells us the values X(ω) for all G-measurable random variables X. In particular, for each event A ∈ G, we know whether A has occurred or not. Example 2.1. Ω = [0, 1] equipped with Borel sets and Lebesgue measure. Let X(ω1, ω2) = ω1. Then for a Borel set A ⊂ [0, 1] X−1A = {(ω1, ω2) : ω1 ∈ A} = A× [0, 1]. hence σ(X) = B([0, 1])× [0, 1]. Example 2.2. Intuitively, If X is a random variable, σ(X) represents the information of knowing X. Let Y = f(X) (meaning Y = f ◦X), where f : R → R is a Borel function. Then Y is σ(X)-measurable. Indeed, {Y ∈ A} = {X ∈ f−1A} ∈ σ(X). One can prove that the converse is also true, if Y is σ(X)-measurable then Y = f(X) for some Borel function f : R → R. Example 2.3. Suppose X1, X2, . . . are random variables over (Ω,F , P ), then