$(X_n)$ is a sequence of $(F_n)$-adapted integrable random variables, where $(F_n)$ is a Filtration and $X_0=0$.
I have to prove that
1)$X_n$ is a martingale with a respect to $F_n$
iff
2)for any finite stopping-time $t$ that takes at most two values, we have $E[X_t]=0$
what i have:
"1) to 2)" we can prove that $X_n$ is uniformly integrable and then use Optional Stopping Theorem to show that $E[X_t]=E[X_0]=0.$
"2) to 1)": we have that $X_n$ is integrable and adapted.
How can i show the Martingale property of $(X_n)$ here, and am i wrong in the first part?
Thanks in advance!
Best Answer
For the implication 2) implies 1):
1. Try first stopping times taking just one value; say, $t=n$ for a positive integer $n$. (Implicit in 2) is the integrability of each random variable $X_n$.)
2. Let $m<n$ be non-negative integers and let $A\in F_m$. Consider the stopping time $$ t(\omega)=\cases{ m,&$\omega\in A$;\cr n,&$\omega\in A^c$.\cr} $$