Stochastic Models for Financial Engineering

A triple where Ω is the sample space of outcomes, 𝓕 is a σ-algebra of measurable events on Ω, and ℙ : 𝓕 → [0, 1] is a countably additive probability measure with ℙ(Ω) = 1.

A collection 𝓕 of subsets of Ω is a σ-algebra if (1) Ω ∈ 𝓕, (2) it is closed under complements (A ∈ 𝓕 ⇒ Aᶜ ∈ 𝓕), and (3) it is closed under countable unions. These three axioms make 𝓕 the natural domain for a probability measure.

An increasing family of σ-algebras 𝓕₀ ⊆ 𝓕₁ ⊆ 𝓕₂ ⊆ ... ⊆ 𝓕, representing the information available up to time t. A process {Xₜ} is adapted if Xₜ is 𝓕ₜ-measurable for every t — i.e., its value is known by time t.

Given an integrable X and sub-σ-algebra 𝓖 ⊆ 𝓕, 𝔼[X | 𝓖] is the (a.s. unique) 𝓖-measurable random variable Y satisfying ∫_A Y dℙ = ∫_A X dℙ for every A ∈ 𝓖. Intuitively, the best L²-projection of X onto information 𝓖.

If 𝓖 ⊆ 𝓗 ⊆ 𝓕, then 𝔼[𝔼[X | 𝓗] | 𝓖] = 𝔼[X | 𝓖]. Conditioning on the coarser information always wins: nested conditioning collapses to the smallest σ-algebra. Special case: 𝔼[𝔼[X | 𝓖]] = 𝔼[X].

Linearity: 𝔼[aX + bY | 𝓖] = a 𝔼[X | 𝓖] + b 𝔼[Y | 𝓖]. Pull-out: if Y is 𝓖-measurable and integrability holds, 𝔼[YX | 𝓖] = Y · 𝔼[X | 𝓖]. Independence: if X ⊥ 𝓖, then 𝔼[X | 𝓖] = 𝔼[X]. Jensen: for convex φ, φ(𝔼[X | 𝓖]) ≤ 𝔼[φ(X) | 𝓖].