Equivalent Conditions for being a Stopping Time

Last updated Nov 1, 2022

Suppose ${\mathcal{B}_{n} : n \in \mathbb{N} }$ is a Discrete-Time Filtration on $\Omega$. Let $\nu : \Omega \to \bar{\mathbb{N}}$. The following are Equivalent:

1. $\nu$ is a Stopping Time.
2. $[\nu \leq n] \in \mathcal{B}_{n}, \forall n \in \mathbb{N}$
3. $[\nu > n] \in \mathcal{B}_{n}, \forall n \in \mathbb{N}$

$(1) \Rightarrow (2)$: We have for $n \in \mathbb{N}$ $$[\nu \leq n] = \bigcup\limits_{m \leq n}[\nu =m] \in \mathcal{B}{n}$$ since $[\nu = m] \in \mathcal{B}{m} \subset \mathcal{B}_{n}$ $\forall m \leq n$. $\checkmark$

$(2) \Rightarrow (3)$: Let $n \in \mathbb{N}$. Since $[\nu \leq n] \in \mathcal{B}{n}$, we know by definition of Sigma Algebra that $[\nu > n] = [\nu \leq n]^{C} \in \mathcal{B}{n}$. $\checkmark$

$(3) \Rightarrow (1)$: Let $n \in \mathbb{N}$. Since $[\nu > n] \in \mathcal{B}{n}$ and because $\mathcal{B}{n}$ forms a Filtration, we have that $$[\nu = n] = [\nu > n-1] \cap ([\nu > n])^{C} \in \mathcal{B}_{n},$$ where $[\nu > 0] = \Omega$. $\checkmark$

$\blacksquare$

1. Condition (3) is the same as saying $[\nu \geq n+1] \in \mathcal{B}_{n}$ $\forall n \in \mathbb{N}$.

1. Resnick - A Probability Path Sect 10.7 pg 365
2. Durret - Probability Theory and Examples pg 220 Sect 4.2

• Function