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$(X_n)$ 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!

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1 Answer 1

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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 $$ t(\omega)=\cases{ m,&$\omega\in A$;\cr n,&$\omega\in A^c$.\cr} $$

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  • $\begingroup$ hi, for first one as you said, we obtain that E[Xn]=0 and therefore Xn is martingale, is that what you mean? $\endgroup$
    – Malik
    Commented Aug 27, 2022 at 20:17
  • $\begingroup$ could you please explain 2. specifically, i can't understand why we define A here $\endgroup$
    – Malik
    Commented Aug 27, 2022 at 20:39
  • $\begingroup$ You get $E[X_0]=0$, which is part of being a martingale (since $X_0=0$). $\endgroup$ Commented Aug 28, 2022 at 19:55
  • $\begingroup$ Using the stopping time $t$ suggested, from $E[X_t]=0$ you deduce that $$ 0=E[X_m\cdot1_{A^c}]+E[X_n\cdot 1_A]=-E[X_m\cdot1_{A}]+E[X_n\cdot 1_A], $$ where the second equality uses $E[X_m]=0$. You have deduced that $$ E[X_n\cdot 1_A]=E[X_m\cdot1_{A}], $$ for each event $A\in F_m$. Refer now to your definition of conditional expectation to see that this implies (indeed it equivalent to) $$ E[X_n\mid F_m]=X_m. $$ $\endgroup$ Commented Aug 28, 2022 at 20:02
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    $\begingroup$ Yes, optional stopping, which is always valid for a two-valued stopping time. $\endgroup$ Commented Aug 30, 2022 at 23:46

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