Stopping Time

Stopping Time

A random variable $\tau :\Omega \to \mathbb{N}\cup \{\infty \}$ is called a stopping time for filtration $({\mathcal{F}}_n{)}_{n\in \mathbb{N}}$ if $\{\omega :\tau (\omega )\le n\}\in {\mathcal{F}}_n$ for each $n\in \mathbb{N}$.

Theorem

If $(M_n{)}_{n\in \mathbb{N}}$ is a martingale and $\tau$ is a stopping time for $({\mathcal{F}}_n{)}_{n\in \mathbb{N}}$, then the stopped process $(M_{n\wedge \tau }{)}_{n\in \mathbb{N}}$ is a martingale.

Proof Without loss of generality, it can be assumed that $M_0=0$, otherwise we could redefine our martingale to be $(M_n-M_0{)}_{n\in \mathbb{N}}.$ Define, for each $k\in \mathbb{N}$, $A_k:=1_{\tau \ge k}=1-1_{\tau \le k-1}.$ Then $(A_k{)}_{k\in \mathbb{N}}$ is a predictable process with respect to $({\mathcal{F}}_k{)}_{k\in \mathbb{N}}$. Now,

\mathbf{1}_{\tau\leq n-1} \sum_{k=1}^n A_k(M_k-M_{k-1}) = \mathbf{1}_{\tau\leq n-1} \sum_{k=1}^{\tau} (M_k-M_{k-1}) = \mathbf{1}_{\tau\leq n-1}M_\tau, $$and$$ \mathbf{1}_{\tau\geq n} \sum_{k=1}^n A_k(M_k-M_{k-1}) = \mathbf{1}_{\tau\geq n} \sum_{k=1}^n (M_k-M_{k-1}) = \mathbf{1}_{\tau\geq n}M_n.

Therefore, $M_{n\wedge \tau }={\sum }_{k=1}^n{A}_k(M_k-M_{k-1}).$ That is, the stopped process is a martingale transform. Theorem then implies that $M_{n\wedge \tau }$ is a martingale.