Given a filtered probability space $(\Omega, \mathscr F, \{\mathscr F_n\}_{n \in \mathbb N}, \mathbb P)$ where $\mathscr F_n = \mathscr F_n^Y$,
let $Y_1, Y_2, ...$ be iid random variables w/ $P(Y_n = 1) = P(Y_n = -1) = 1/2$.
$X$ is a symmetric random walk: $X_0 = 0, X_n = Y_1 + ... + Y_n$ and is a $(\{\mathscr F_n\}_{n \in \mathbb N}, \mathbb P)$-martingale.
It can be shown that $S = \inf\{n : X_n = 7\} < \infty$ and $T = 10^{12} \wedge S$ are $\{\mathscr F_n\}$-stopping times.
Find $E[X_S]$ and $E[X_T]$.
What I tried:
$$E[X_S] = E[\lim X_{S \wedge n}]$$
$$\stackrel{DCT}{=} \lim E[X_{S \wedge n}] \tag{*}$$
where
$$X_{S\wedge n} = ( \ \sum_{k=0}^{n} 1_{\{S=k\}} X_k \ ) + 1_{S> n} X_n$$
$$\to E[X_{S\wedge n}] = E[( \ \sum_{k=0}^{n} 1_{\{S=k\}} X_k \ ) + 1_{S> n} X_n]$$
$$= ( \ \sum_{k=0}^{n} E[1_{\{S=k\}} X_k] \ ) + E[1_{S> n} X_n]$$
$$= ( \ \sum_{k=0}^{n} E[1_{\{S=k\}} 7] \ ) + E[1_{S> n} X_n]$$
$$= ( \ \sum_{k=0}^{n} 7P(S=k) \ ) + E[1_{S> n} X_n]$$
Now
$$E[1_{S= n+1} X_n] = E[X_n | S= n+1] P(S= n+1) = E[X_n] P(S= n+1) = 7 P(S= n+1)$$
Similarly, $E[1_{S=k} X_n] = 7 P(S= k)$ for $k > n+1$.
It seems
$$E[X_S] = E[X_{S \wedge n}] = 7 = E[X_{S\wedge 10^{12}}]$$
Is that right?
$(*)$ DCT
$$X_{S\wedge n} = ( \ \sum_{k=0}^{n-1} 1_{\{S=k\}} X_k \ ) + 1_{S\ge n} X_n$$
$$|X_{S\wedge n}| = |( \ \sum_{k=0}^{n-1} 1_{\{S=k\}} X_k \ ) + 1_{S\ge n} X_n|$$
$$\le |( \ \sum_{k=0}^{n-1} 1_{\{S=k\}} X_k \ )| + |1_{S\ge n} X_n|$$
$$\le ( \ \sum_{k=0}^{n-1} |1_{\{S=k\}} X_k| \ ) + |1_{S\ge n} X_n|$$
$$\le ( \ \sum_{k=0}^{n-1} 1_{\{S=k\}} |X_k| \ ) + 1_{S\ge n} |X_n|$$
$$\le ( \ \sum_{k=0}^{n-1} |X_k| \ ) + |X_n|$$
$$\le \sum_{k=0}^{n} |X_n|$$
$\because X$ is a martingale, $E[|X_n|] < \infty$
$$\to E[\sum_{k=0}^{n} |X_n|] < \infty $$
