Limit of monotone sequence of stopping times.

Question

I am wondering if the next statement is true:

Let $\{\tau_n\}$ a non-increasing sequence of stopping times, then if we define $\tau_0=\lim_{n\rightarrow\infty}\tau_n$ is also a stopping time.

I think is true but I am not sure. Thanks for any help.

Answer 1

By definition, $\eta$ is a stopping time, if $\{\eta\le t\}\in\mathcal{F}_t$ for all $t\in T$ where $T$ is the set of times; equivalently, $\{\eta> t\}\in\mathcal{F}_t$. Since your sequence is non-increasing $\lim_n \tau_n=\inf_n \tau_n$, and assuming your times are discrete (subset of $\mathbb{Z}$), we have $$ \{\lim_n \tau_n>t\}=\{\inf_n \tau_n>t\}=\{\tau_n>t\text{ for all }n\}=\bigcap_n \{\tau_n>t\}\in\mathcal{F}_t, $$ since each event in the intersection belongs to $\mathcal{F}_t$, so the answer is "yes". In case of continuos time, there can be some subtleties.

I think in case of continuous time this is not true. Consider the following process X(t). From time zero until some random time $\eta$ it stays equals to -1. At this time it jumps to zero in such a way that the process is left continuous, that is, equals -1 at time $\eta$ itself. After that, X is just a Brownian motion. Define $\tau_n=\inf\{t: X(t)\ge 1/n\}$. Then the limit of them is exactly $\eta$, which is not a stopping time due to the fact the process is not right continuous at this point.