Filtrations

Why

Definition

Let $(I, \leq)$ be a totally ordered set. A family of sigma-algebras $\set{F_i}_{i \in I}$ is a filtration if $F_{j} \subset F_{k}$ for all $j \leq k$.

We call a filtration whose index set is the natural numbers with their usual order a discrete-time filtration. We call a filtration whose index set is the real numbers with their usual order a continuous-time filtration. The index set may also be finite, for example, $\set{F_i}_{i = 1}^{n}$, in which case

\[ F_1 \subset F_2 \subset \cdots \subset F_n. \]

Notation

It is extremely common to see filtrations written with the caligraphic $\mathcal{F} $. As in, let $\set{\mathcal{F} _{i}}_{i=1}^{n}$ be a filtration. This is in accordance with using caligraphic letters for sets of sets, and in accordance with the term sigma field for sigma algebra.

1. Future editions will include, and likely will need stochastic processes. ↩︎