Trouble with extending Doob's Optional Stopping Theorem

Let $\tau\geq0$ be a stopping time, $\mathbb{E}\tau<\infty$.

• Show $\{\tau\geq k\}\in\mathcal{F}_{k-1}$.
• Based on the identity

$$ |X_{T\land n}|= \bigg|\sum_{k=1}^n(X_k-X_{k-1})\cdot\mathbf{1}\{\tau\geq k\}\bigg|\leq\sum_{k=1}^n|X_k-X_{k-1}|\cdot\mathbf{1}\{\tau\geq k\} $$ show the following:

If $X$ is a supermartingale for which there exists a $C\in\mathbb{R}$ with $$ \mathbb{E}(|X_k-X_{k-1}|\big|\mathcal{F}_{k-1})\leq C\quad\forall k>0,\text{ a.s.}, $$ then $\mathbb{E}X_\tau\leq\mathbb{E}X_0$. Of course we have equality in case $X$ is a martingale.

The first part is easy, but I am struggling with understanding the solution to the second part.

The solution it gives for the second part goes as follows: $$ \begin{align*} \mathbb{E}\sum_{k=1}^\infty\big(|X_k-X_{k-1}|\cdot\mathbf{1}\{\tau\geq k\}\big)&=\sum_{k=1}^\infty\mathbb{E}\big(|X_k-X_{k-1}|\cdot\mathbf{1}\{\tau\geq k\}\big)\\ &=\sum_{k=1}^\infty\mathbb{E}\mathbb{E}\big(|X_k-X_{k-1}|\cdot\mathbf{1}\{\tau\geq k\}\big|\mathcal{F}_{k-1}\big)\\ &=\sum_{k=1}^\infty\mathbb{E}\big[\mathbf{1}\{\tau\geq k\}\cdot\mathbb{E}\big(|X_k-X_{k-1}|\big|\mathcal{F}_{k-1}\big]\\ &\leq C\sum_{k=1}^\infty\mathbb{E}\mathbf{1}\{\tau\geq k\}=C\sum_{k=1}^\infty\mathbb{P}\{\tau\geq k\}=C\mathbb{E}\tau<\infty. \end{align*} $$ So far so good. But then the next step in the solution confuses me. It says that we can now apply dominated convergence and a.s. finiteness of $\tau$ to claim $$ 0\geq\lim_{n\to\infty}\mathbb{E}(X_{T\land n}-X_0)=\mathbb{E}X_\tau-\mathbb{E}X_0. $$ However, what confuses me, is that I thought to apply dominated convergence we needed to show that the variable itself could be bounded, not just the expectation of the variable. I.e. the statement of the dominated convergence theorem I have is:

Let $Y,X,X_1,X_2,X_3,\dots$ be random varialbes, and assume $|X_n|\leq Y$ for each $n$, $\mathbb{E}Y<\infty$, and $X_n\to X$ almost surely. Then $\mathbb{E}X_n\to\mathbb{E}X$. It seems that this statement does not apply directly to this question. What am I missing here that has allowed the use of dominated convergence in this situation? Any help is sincerely appreciated!

Solution 1:

DCT can be applied to sequences $(u_n)_{n \in \mathbb{N}}$ of measurable functions s.t. $|u_n|\leq w \in \mathcal{L}^1,\,\forall n$. In this case define $$w(\omega)=\sum_{k \in \mathbb{N}}|X_k(\omega)-X_{k-1}(\omega)|\mathbf{1}_{\{\tau\geq k\}}(\omega)$$ we have shown that $0\leq E[w]\leq CE[\tau] <\infty$. Thus $w \in \mathcal{L}^1$. Since $|u_n|=|X_{T \wedge n}|\leq w,\,\forall n$ and $X_{T \wedge n}\to X_T$ (a.e.) then we can apply DCT.