Let $X_t$ be a Levy process, with its associated running supremum $$ \bar X_t = \sup_{s \le t} X_s. $$ With $w >0$, define the process $$ W_t = \left( w \lor \bar X_t \right) - X_t. $$ Show that, on the event $\{W_t=0\}$, $$ dt=dX_t=d\bar X_t, \tag{1} $$ then obtain that $$ \int_0^t \mathbf 1_{\{W_s =0\}} ds = \int_o^t \mathbf 1_{\{X_s=\bar X_s, X_s > w\}} d\bar X_s. \tag{2} $$
Assuming (1) holds, I can get (2).
For (1), I know
$$
\{W_t=0\} = \{X_t = \bar X_t, \bar X_t > w \}. \tag{3}
$$
I don't see how (3) gives information on the increment $dt$.
This is related to this question.
