Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

This is a series problem and I'm struggling with the last part. I assume that the last part has nothing to do with previous ones, so i won't put up the other parts. Question is :

Let $\tau$ be a stopping time and $X=(X_n,{\cal F_n})$ be a martingale. Show that stopped sequence $X_{n}^{\tau}=( X_{\tau_{n}},{\cal F_n})$ where $\tau_{n}=\min\{\tau,n\}$ is a martingale.

I started with $E(X_{\tau_{n}}|F_{n-1})=E(X_{\min\{\tau,n\}}|F_{n-1})$ which then would have two cases, but all ends up with $E(X_{n-1})$, or should that be $E(X_{\tau_{n-1}})$?

share|cite|improve this question
Neither. The result should be $X_{\tau_{n-1}}$. // You could decompose the conditional expectation along the events $[\tau\leqslant n-1]$ and $[\tau\geqslant n]$ and compute the two terms separately and you might first try to show that $[\tau\geqslant n]$ belongs to $F_{n-1}$. – Did Nov 4 '11 at 8:26
kkk: Any luck with these hints? – Did Nov 5 '11 at 11:24

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.