Quadratic Variation of Increasing Process? I am looking through my notes and I came across the following statement: 
Let $X_s$ be a positive local martingale and let $M_t = max_{0 \le s \le t} X_s$. Then since $M_t$ is an increasing process, $[X,M] = [M] = 0$. Why is this true?
On a similar note, the claim is made that in terms of stochastic integrals, $dM_s$ is $0$ unless $X_s = M_s$. I'm not sure why this holds either.
 A: Let $(\pi_n)_{n\geq 0}$ be a sequence of partitions of $[0,T]$ whose mesh is going to zero (that is, if the partition is $\pi_i:0=t_0\leq t_1 \leq ... t_{N(i)} = T$, then $mesh(\pi_i)=max_p(t_{p+1}-t_p)$).
Then a/the definition of quadratic covariation of processes X, Y over $[0,T]$ is:
$$
[X,Y]_T = lim_{n \to \infty} \sum_{i=0}^{N(n)-1}|X_{t_{i+1}}-X_{t_i}||Y_{t_{i+1}}-Y_{t_i}|
$$
(see Revuz & Yor, Continuous Martingales and Stochastic Calculus, Chapter IV on Stochastic Integration, Theorem 1.9 for the equivalence of this definition with the other main definition; that quadratic covariation is the unique increasing stochastic process such that $(X_tY_t-[X,Y]_t)_{t\geq 0}$ is a martingale) and $[X]:=[X,X]$.
Then if M is some general increasing process, and X some general continuous process, then we prove that $[X,M]_T=0$ for all T:
$$
[X,M]_T = lim_{n \to \infty} \sum_{i=0}^{N(n)-1}|X_{t_{i+1}}-X_{t_i}||M_{t_{i+1}}-M_{t_i}| \\
\leq lim_{n \to \infty} \sum_{i=0}^{N(n)-1}(max_j|X_{t_{j+1}}-X_{t_j}|)|M_{t_{i+1}}-M_{t_i}| \\
= lim_{n \to \infty} (max_j|X_{t_{j+1}}-X_{t_j}|)\sum_{i=0}^{N(n)-1}|M_{t_{i+1}}-M_{t_i}|\\
= lim_{n \to \infty} (max_j|X_{t_{j+1}}-X_{t_j}|)\sum_{i=0}^{N(n)-1}(M_{t_{i+1}}-M_{t_i})
$$
since M is increasing, and then as the sum is telescoping:
$$
= lim_{n \to \infty} (max_j|X_{t_{j+1}}-X_{t_j}|)(M_T-M_0)=0
$$
since X is continuous, and the mesh of the partition is tending to zero, so $max_j|X_{t_{j+1}}-X_{t_j}|\to 0$
Since our proof here is for general processes X, M, with M increasing and X increasing, we thus have $[X,M]=[M](=[M,M])=0$.
This will not in general be true if X is not a continuous process. If X has a jump discontinuity which causes M to have a jump discontinuity, say at $t\in (t_i, t_{i+1})$, then $|X_{t_{i+1}}-X_{t_i}||M_{t_{i+1}}-M_{t_i}|$ can never go to zero; it will always equal at least the product of the two jumps. For a partial discussion on this see https://almostsuremath.com/2010/01/19/properties-of-quadratic-variations/.
As for your second question, imagine drawing sample paths for X and M on the same graph. So at every point, M is either constant or increasing. But by the definition of M, M can only increase, when X is reaching a new, higher maximum. But, whenever X is reaching a new maximum, say at $X_t$, $M_t = X_t$ by definition of M being the maximum X up to that point. Thus the only case when $dM_s=0$ is when $M_s=X_s$. I hope that makes sense - try drawing the sample paths of X and M for yourself; this should elucidate what I'm trying to explain.
