0

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.

Don P.
  • 313
  • What have you tried? – geetha290krm Feb 09 '23 at 04:58
  • GIven that ${\tau_n}$ is a non-increasing sequence of stopping times bounded below (by $0$) $\tau_0=\lim_{n\rightarrow\infty}\tau_n$ is well defined. And I have that $[\tau_0\leq t]\subset\bigcup_{n=1}^\infty[\tau_n\leq t]\in\mathcal{F}_t$, but I am not sure if the other contention is true. – Don P. Feb 10 '23 at 00:09
  • By the below comment of @Stas $[\tau_0>t]=[\inf_{n}\tau_n>t]=\bigcap_{n=1}^\infty[\tau_n>t]\in\mathcal{F}_t$. – Don P. Feb 10 '23 at 00:15

1 Answers1

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.

van der Wolf
  • 2,054
  • Thank you very much for your answer, but I can't understand el "why" should have to be discrete, I don't see how this affect your arguments. And I think you have a little typo in "non-decreasing", it should be "non-increasing", right? – Don P. Feb 09 '23 at 23:51
  • 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. – van der Wolf Feb 10 '23 at 07:18
  • 1
    Thank you very much for your answer, I realised that the problem in continuous time is $\bigcap_n[\tau_n>t]=\left[\lim_n\tau_n\geq t\right]$ but this only implies that $[\lim_n\tau_n<t]\in\mathcal{F}_t$, now it is clear, thanks again – Don P. Feb 10 '23 at 17:29