Originally Posted by

**akbar** Let

be a uniformly integrable martingale,

. Let

and

(

if

for all

). One can check

is a stopping time of

.

How do you prove that

is also a uniformly integrable martingale?

You agree it suffices to prove ? (since given the definition of )

We have , and for all , and in fact we even have where is the almost-sure (and L1) limit of using the classical theorem about uniformly integrable martingales. Thus , hence:

(using the stopping time definition) and thus finally:

.

In fact, for any stopping time , the stopped martingale is still uniformly integrable. You can deduce a proof from Theorem 12.5.4 (remark (i)) in these lecture notes.