We work on a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\in [0,T]},P)$. Let $X$ be an adapted làdlàg stochastic process (i.e. the left and right limits exist). For $0<t<T$ denote by $\Delta_+ X_t := X_{t+}-X_t$ its right and by $\Delta X_t := X_t -X_{t-}$ its left jumps.
In this paper on page 1894 it is claimed that the continuous part of $X$ is given by \begin{equation} X_t^c:=X_t -\sum_{s<t}\Delta_+ X_s -\sum_{s\leq t}\Delta X_s. \end{equation} However, since $\sum_{s<\cdot}\Delta_+ X_s$ is right-continuous, shouldn't we take the left limit of this sum, i.e. \begin{equation} X_t^c:=X_t -\left(\sum_{s<t}\Delta_+ X_s\right)_- -\sum_{s\leq t}\Delta X_s \end{equation} so that $X^c$ is continuous?