Martingale with the stopping time $T=\inf\{n\geq 1 \, : \, |S_n| \leq c\}$

Question

Suppose that $X_i$, $i \geq 1$ are independent random variables with $\mathbb{E} X_i = 0$ and $\text{var}(X_i) = \sigma_i ^2 < \infty$.

Assume that $X_i$ are uniformly bounded, i.e., there exists $C>0$ such that $|X_i(\omega)| < C $ for all $i\geq 1$ and $\omega \in \Omega$.

Let $c>0$ and $T=\inf\{n\geq 1 \, : \, |S_n| \geq c\}.$

Claim is to show that $\mathbb{E} (S_{T \wedge n} ^2) \leq (C+c)^2$, where $S_n = X_1 + \cdots + X_n$.

It looks very simple but I failed to prove it.

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I tried to use the predictable sequence $H_n = \mathbb{1}_{\{T\geq n\}}$ because $$S_{T\wedge n}^2 = \sum_{m=1}^n H_m (S_m^2 - S_{m-1}^2),$$ but it didn't work.

Any help will be appreciated!

Answer 1

$$ E [ S_{n \wedge T}^2 ] = E [ S_n^2 , n <T] + E [S_T^2, T \le n].$$

1. On $\{n<T\}$, $|S_n|<c \le (C+c)$.
2. On $\{T\le n\}$, either $T=1$, in which case clearly $|S_T|\le C \le (C+c)$, or $T>1$ in which case, the definition of $T$ implies: $$|S_T| = |S_{T-1} +X_{T-1}|\le |S_{T-1}|+|X_{T-1}|\le c + C.$$

Summarizing,

$$ E [S_{n\wedge T}^2 ] \le (C+c)^2 P(n<T) + (C+c)^2 P(T\le n) = (C+c)^2.$$