Discrete-Time Filtration

Last updated Nov 1, 2022

Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a Probability Space. The Collection $\mathcal{F} = {\mathcal{B}_{n} : n \in \mathbb{N}}$ is a Discrete-Time Filtration if $\mathcal{F}$ is a Filtration wrt Index Set $\mathbb{N}$.

1. It is sometimes convenient to extend $\mathcal{F}$ to a Filtration over $\bar{\mathbb{N}}$ by defining $$\mathcal{B}{\infty} = \sigma(\mathcal{B}{n} : n \in \mathbb{N}).$$

1. Natural Numbers
2. Extended Natural Numbers