what is filtration in local martingales theory?

Question

according to wiki Let $(Ω, F, P)$ be a probability space; let $F_∗ = { F_{t | t ≥ 0} }$ be a filtration of$ F$; let$ X : [0, +∞) × Ω → S $ be an $F_∗ $ -adapted stochastic process on set S. Then X is called an $F_∗$ -local martingale if there exists a sequence of $F_∗$-stopping times $τk : $Ω → [0, +∞)$. what is is this filtration .can somebody explain ?

Answer 1

A filtration $(F_t)_{t \geq 0}$ on $(\Omega,F)$ is a family of $\sigma$-algebras satisfying

• $F_t \subseteq F$ for each $t \geq 0$
• $F_s \subseteq F_t$ for any $s \leq t$.

One of the most common filtrations is the canonical filtration of a stochastic process; that is $$F_t := \sigma(X_s; s \leq t).$$ (That is, $F_t$ is the smallest $\sigma$-algebra on $\Omega$ such that the mappings $\omega \mapsto X_s(\omega)$ are measurable for all $s \leq t$.)