Let $X_i$ be measurable for the $\sigma$-field $F_i$. Suppose that for some constants $a_i, c_i \in \mathbb{R}$, $$ \mathbb{E}\left(X_i-X_{i-1} \mid F_{i-1}\right)<a_i \quad \text { and } \quad\left|X_i-X_{i-1}-a_i\right|<b_i \quad \text { a.s. } $$ We want to show that for $t > 0$, $$ \mathbb{P}\left(\max _{i \in[n]} X_i>X_0+\sum_{i=1}^n a_i+t\right)<\exp \left(-\frac{t^2}{2 \sum_{i = 1}^n b_i^2}\right). $$
I think this is something like Azuma's inequality, we can let $S_n = \sum_{i = 1}^n (X_i - X_{i - 1})$, then we can apply Azuma's inequality to show that $$ \mathbb{P}\left( X_n - X_0 > \sum_{i=1}^n a_i+t\right)<\exp \left(-\frac{t^2}{2 \sum_{i = 1}^n b_i^2}\right). $$ However, I do not know how to deal with $\max_{i \in [n]} X_i$.