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Filtration

Last updated Nov 1, 2022

# Definition

Let (Ω,A,P)(\Omega, \mathcal{A}, \mathbb{P}) be a Probability Space

, let (I,)(I, \leq) be a Total Ordering

Total Ordering

Definition A (T,)(T, \leq) is a if x,yT\forall x,y \in T, either xyx \leq y or yxy \leq x....

11/7/2022

. Let $\mathcal{B}{i} \subset \mathcal{A}$ be Sub-Sigma Algebra

Sub-Sigma Algebra

Definition A is a subset of a that is also a ....

11/7/2022

s of A\mathcal{A}. Then $(\mathcal{B}
{i}){i \in I}$ is a Filtration

Filtration

Definition Let (Ω,A,P)(\Omega, \mathcal{A}, \mathbb{P}) be a , let (I,)(I, \leq) be a . Let BiA\mathcal{B}{i} \subset \mathcal{A} be s of...

11/7/2022

if $\mathcal{B}
{k} \subset \mathcal{B}_{l}when when k \leq l$.

That is, the Function

Function

Definition Let X,YX, Y be s. Then a XX to YY is a fX×Yf \subset X \times Y such...

11/7/2022

$f: I \to {\mathcal{B}{i}: i \in I}definedby defined by f(i) = \mathcal{B}{i}$ Non-Decreasing Function.