# The inequality of Submartingale and Stopping time

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

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

$$|X_n|$$ 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 $$\lim E|X_n|$$.