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First there have theorem

\begin{aligned} &\text { Theorem 5.4.1. If } X_{n} \text { is a submartingale and } N \text { is a stopping time with } \mathbb{P}(N \leq k)=1\\ &\text { then } \mathbb{E} X_{0} \leq \mathbb{E} X_{N} \leq \mathbb{E} X_{k} \end{aligned}

but this theorem is related to bounded stopping time
Then there have an exercise

Let $ \left \{X_ {n}, \mathcal{F}_{n} \right \}_{n \geq 0} - $ martingale or nonnegative submartingale, and let $ \tau- $ stop time with respect to $ \left \{\mathcal{F}_{n} \right \}_{n \geq 0} $ such that $ \mathrm{P} (\tau <\infty) = 1. $

How to prove that $ \mathrm {E} \left | X_{\tau} \right | \leq \lim_{n \rightarrow \infty} \mathrm{E} \left | X_{n} \right | $

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  • $\begingroup$ I think considering $X_{\min\{\tau, n\}}$ might be helpful. $\endgroup$
    – angryavian
    Mar 5, 2021 at 5:09

1 Answer 1

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$|X_n|$ is a non-negative submartingale in both cases.

$E|X_{\min \{\tau,n\}}|\leq E|X_n|$ by Theorem 5.4.1 and Fatou's Lemma gives $E|X_{\tau}| =E \lim |X_{\min \{\tau,n\}}|\leq \lim \inf E|X_{\min \{\tau,n\}}| \leq \lim \inf E|X_n| $. Since $(E|X_n|)$ is monotonic its limit exists so $\lim \inf E|X_n| $ is same as $\lim E|X_n|$.

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