Question about a predictable process Q) If $X_n$ is a submartingale and $M\leq N$ are stopping times with $P(N\leq k)=1$, then show that $EX_M\leq EX_N$.
Ans) Let $H_n = 1_{\{M<n\leq N\}}$. I can see that $H_n$ is a predictable process. Then 
$$(H.X)_n = X_{n\wedge N}-X_{n\wedge M} \tag{1}$$
I don't understand how we get the above equation. $(H.X)_n$, in general, is defined as:
$$(H.X)_n = \sum_{m=1}^nH_m(X_m - X_{m-1}) $$
I mean if $M<m=1,...,n<N$, then $(1)$ checks out but I get confused with indicators, so may I know how we get $(1)$ in general?
If $(1)$ is true, then $X_{n\wedge N}-X_{n\wedge M}$ is also a submartingale and taking $E$ at $n=0$ and $n=k$ gives the answer which I understand.
 A: Note the definition $(H\cdot X)_n = \sum_{m=1}^nH_m(X_m - X_{m-1})$, and recall the fact that $H_m \in \mathcal{F}_{m-1}$. In the context of $(H\cdot X)_n$, one can interpret $H_m$ as a coefficient to be decided at time $m-1$ for preserving the upcoming increment $X_m - X_{m-1}$. 
Defining $H_m$ can be understood as a mechanism for constructing new (super/sub)martingales out of a given one by using the increments/differences of the given (super/sub)martingale.
Following the line of thinking, suppose you want to construct the sequence $X_{n\wedge N} - X_{n\wedge M}$. Your first analyze the target sequence to understand that : (1) it is 0 for $n \le M$ (2) it is $X_n - X_M$ for $M < n \le N$ (3) it stays unchanged at $X_N$ - $X_M$ for $n > N$. Then you can construct the sequence by accumulating all the differences between $X_M$ and $X_N$ (and set $H_m=0$ for $m>N$ or $m \le M$). Hence you define $H_n = 1_{\{M<n\leq N\}}$ to mean you want to accumulate increments $X_m - X_{m-1}$ for $M < m \le N$ with coefficients being $1$.
Another simpler example of constructing $X_{n\wedge T}$ from $X_n$:
$X_{n\wedge T}=X_n$ for $n \le T$, and $X_{n\wedge T}=X_T$ for $n\ge T$. Hence you would want to accumulate all increments up to time $T$ to keep the sequence unchanged. And you set the coefficients to zero after $T$. Hence $H_m=1_{(m\le T)}$. Then $(H\cdot X)_n = X_{n\wedge T}$.
When $H_m$ is defined as an indicator function as $H_{(L\le n \le U)}$, the lower bound $L$ should be the index of the first non-zero items in the new sequence, and the upper bound $U$ should be the index of the last item that is different from its predecessor. Then $(H\cdot X)_n$ in between are the original sequence values (with an offset of $-X_{L-1}$).
Hope it helps.
