Are these true for a martingale? $E [\frac{X_{n + 1}}{X_{n}}] = 1, E [\frac{X_{n + 2}}{X_{n}}] = 1$

Question

Let $(X_{n})_{n \in Z_{+}}$ be a martingale, $X_{n} (ω) \neq 0$ , and $X_{n + 1} / X_{n}, X_{n + 2} / X_{n} \in L^{1} (n \in Z_{+})$

Do the following hold for $n \in Z_{+}$ ?

$E [\frac{X_{n + 1}}{X_{n}}] = 1, E [\frac{X_{n + 2}}{X_{n}}] = 1$

Kavi Rama Murthy · Accepted Answer · 2020-07-24 05:50:32Z

2

$E (X_{n + m} | F_{m}) = X_{m}$ . This implies $E (\frac{X_{n + m}}{X_{m}} | F_{m}) = 1$ provided $\frac{X_{n + m}}{X_{m}}$ is integrable. Taking expectation on both sides we get $E (\frac{X_{n + m}}{X_{m}}) = 1$ .

answered Jul 24 '20 at 5:50

Kavi Rama Murthy

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So you're using that $\frac{1}{X_{m}}$ is measurable in $F_{m}$ , right? – Kay Jul 24 '20 at 5:55
@Kay Yes, exactly. – Kavi Rama Murthy Jul 24 '20 at 5:56
Thank you for the help! – Kay Jul 24 '20 at 5:57

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Are these true for a martingale? $E [\frac{X_{n + 1}}{X_{n}}] = 1, E [\frac{X_{n + 2}}{X_{n}}] = 1$

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