One Martingale problem

Question

In the setting of Kolmogorov's maximal inequality, I need to prove the following

$$P(\max_{1\leq m \leq n}|S_m| \leq x) \leq \frac{(x+K)^2}{var(S_n)}$$

Hint: Use the fact that $S_n^2 -s_n^2$ is a Martingale.

No idea how to proceed.

Answer 1

You're missing an additional condition: $S_n=\sum_{k=1}^nX_k$ with $\mathbb{E}X_k=0, |X_k|\le K$.

Let $s_n^2=\sum_{k=1}^n \mathbb{E}X_k^2$ and $T=\inf\{k:|S_k|>x \text{ or } k=n\}$ ($T$ is a stopping time). Since $S_n^2-s_n^2$ is a MTG and wp1 $T\le n$

$$0=\mathbb{E}[S_T^2-s_T^2]\le (x+K)^2\cdot P\{\max_{1\le k\le n}|S_k|>x\}+(x^2-var(S_n)) \cdot P\{\max_{1\le k\le n}|S_k|\le x\}$$

(the inequality follows from the fact that on $\{\max_{1\le k\le n}|S_k|>x\}$, $S_T^2\le (x+K)^2$ and on $\{\max_{1\le k\le n}|S_k|\le x\}$, $S_T^2=S_n^2\le x^2$).

Hence, $$P\{\max_{1\le k\le n}|X_k|\le x\}\le \frac{(x+K)^2}{var(S_n)}$$