# $dt=dX_t=d\bar X_t$ on the event $\{X_t = \bar X_t, \bar X_t > w \}$, $\bar X$ running supremum of $X$

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 $dt$.

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