Let $X\_{t}$ be a semimartingale. Define 
 $\Delta X\_{t} = X\_{t}- X\_{t-}$.
For fixed $s> 0$, $\Delta X\_{s}$ and $X\_{s-}$ are two random variable. Are they independent to each other? I think the answer is yes. But I am not sure the proof. 



The jump and the left martingale of semimartingale
Let $W\_t$ be 1-dimensional Brownian motion and let
$$V\_t=W\_t+\sum\_{n\in\mathbb N,\, n\le t} W\_n$$
Then
\begin{eqnarray}
\Delta V\_n&=&W\_n,\quad\text{whereas}\\
V\_{n-}&=&\sum\_{m\in\mathbb N,\, m\le n} W\_m
\end{eqnarray}
so $\Delta V\_n$ and $V\_{n-}$ are not independent of eachother.


