Problem on martingale concentration inequality Let $M_n$ be a $F_n$ measurable martingale where $F_n$ are family of increasing sub sigma fields. Let $D_n = M_{n}-M_{n-1}$, for $n\geq 2$. Let $B \in F_1$ and on the set $B$, $a_k \leq D_k\leq b_k$. Then I have to show that
for all $t > 0$
$$P\left(\max_{1\leq k \leq n}|M_k| \ge t \mid B\right) \leq 2\exp{\left(\frac{-2t^2}{\sum_{k\leq n}(b_k -a_k)^2}\right)}$$
I have tried to replicate through the Azuma Hoeffding inequality proof (https://matthewhr.wordpress.com/2012/12/06/azuma-hoeffding-inequality-2/) with the probability measure $P_B = P(A\cap B)/P(B)$, but stuck as with respect to this new measure the original martingale does not become a martingale.
 A: Let us denote by $\mathbb E_B$ the (conditional) expectation with respect to $\mathbb P_B$. 
We have for each $k\geqslant 2$ and each $G\in \mathcal F_{k-1}$, 
\begin{align}
\mathbb E_B\left[\mathbf 1_G\mathbb E_B\left[M_k\mid\mathcal F_{k-1}\right]\right]&=\mathbb E_B\left[\mathbf 1_GM_k\right] &\mbox{by definition of conditional expectation}\\
&=\mathbb E\left[\mathbf 1_{B\cap G}M_k\right]/\mathbb P(B)&\mbox{by definition of }\mathbb P_B\\
&=\mathbb E\left[\mathbf 1_{B\cap G}\mathbb E\left[M_k\mid\mathcal F_{k-1}\right]\right]/\mathbb P(B)&\mbox{because }B\in \mathcal F_1\subset\mathcal F_{k-1}\mbox{ hence }B\cap G\in \mathcal F_{k-1}\\
&=\mathbb E\left[\mathbf 1_{B\cap G}M_{k-1}\right]/\mathbb P(B)&\mbox{because }\left(M_n,\mathcal F_n\right)_{n\geqslant 1} \mbox{ is a martingale for }\mathbb P\\
&= \mathbb E_B\left[\mathbf 1_{G}M_{k-1}\right]&\mbox{by definition of }\mathbb P_B.
\end{align}
This prove that $\mathbb E_B\left[M_k\mid\mathcal F_{k-1}\right]=M_{k-1}$, hence we can apply the non-conditional Azuma-Hoeffding inequality.
