Quadratic variation of supremum of brownian motion

Question

I would like to know if in some book or how could I compute the quadratic variation of the supremum of the bronian motion $S_t=\sup_{s\in[0,t]}W_s$ where $W$ is a Brownian motion. I was thinking about using the fact that $S_t\overset{d}{=}|W_t|$ but also I don't know if two martingales with the same distribution has the same quadratic variation (I know that the converse is not true). Anyone could recomend some lectures about these subjects please, or give some advice about how to start the proofs I'll really appreciate it. Thanks.

Answer 1

The quadratic variation is identically $0$, i.e.

$$\langle S, S \rangle_t = 0$$

for all $t$, almost surely.

To see this, note that $S$ is almost surely increasing, hence has bounded variation almost surely. Thus the quadratic variation is simply equal to the sum of the squared jumps of the process. However, $S$ is also almost surely continuous, so the quadratic variation is identically $0$.

Some comments are in order:

1. The formula you gave for $S$ in terms of $|W|$ does not hold. The correct equivalence is $D_t := S_t - W_t \overset{d}{=} |W_t|$. For $S$ itself, we have that $S_t \overset{d}{=} 2 L_t(0)$, where $L_t (0)$ denotes the local time of $W$ at $0$.

2. Neither $S$ nor $D$ are martingales. However, $S$ is almost surely increasing, hence trivially a submartingale. On the other hand $D$ is a semimartingale, with semimartingale decomposition given by Tanaka’s formula applied to $|W_t|$. Some justification is required as to why two processes with the same law admit the same (in law) semimartingale decomposition.