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.
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
2
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.
answered Feb 25, 2014 at 1:10
-
$\begingroup$ Levy processes would be different $\endgroup$ Commented Feb 25, 2014 at 2:30
-
1$\begingroup$ I got this. In the case of Levy process it is true. More generally, this conclusion is ture for additive process. We only need the increment $X_{t}-X_{s}$ is independent of $X_{s}$. $\endgroup$ Commented Feb 25, 2014 at 20:13