Why do we need σ-algebras? Conditions for a σ-algebra?
To define which sets are “measurable” so probabilities or measures can be assigned consistently.
- Ω ∈ 𝓕
- If A ∈ 𝓕, then Aᶜ ∈ 𝓕
- If A₁, A₂, … ∈ 𝓕, then ⋃ₙ Aₙ ∈ 𝓕
What does σ(A) mean?
Why is σ(A) called the “smallest” σ-algebra?
The smallest σ-algebra containing the set A (or collection A).
It is the intersection of all σ-algebras that contain A.
What is an open set in ℝ?
What is the Borel σ-algebra on ℝ?
A set where every point has an ε-neighborhood fully contained in the set.
The σ-algebra generated by all open sets in ℝ. Denoted by 𝓑(ℝ). Eg: [a, b], (a, b], {x}.
What is the expected value of an Itô integral? State the Itô isometry.
Zero: E[∫₀ᵗ X_s dW_s] = 0.
E[(∫₀ᵗ X_s dW_s)²] = E[∫₀ᵗ X_s² ds].
Compute ∫₀ᵗ c dW_s where c is constant.
Evaluate ∫₀ᵗ W_s dW_s.
cW_t.
½ (W_t² − t).
