just wondering, why $\Delta[Var(X)]_t=|\Delta X_t|$ holds for cadlag, adapted and increasing real valued processes X with finite variation over all finite intervals $[0,t]$, which are in addition uniform integrable martingales? $Var(X)_t$ denotes the variation of a process $X$ over the interval $[0,t]$. Any help?
It is mentioned in the proof of Lemma 3.11 in the book Limit theorems for stochastic processes by Shyraev/Jacod.