simultaneous jumps of independent Levy processes

Suppose I have two independent Levy processes $X_t$ and $Y_t$, both not continuous.

Is anyone familiar and can refer me to a result(or a counterexample) which states that

${\displaystyle \sum_{0\leq s\leq t}}|\bigtriangleup X_{s}(\omega)\bigtriangleup Y_{s}(\omega)|=0$ for all $t\in \mathbb{R}$ a.s?

A different yet equivalent formulation of this is

$\bigtriangleup X_{t}=0$ or $\bigtriangleup Y_{t}=0$ a.s. for all $t\in \mathbb{R}$

In words, every two independent Levy processes have no simultaneous jumps a.s. I know it holds for independent Poisson processes and I'm wondering if it generalizes.

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You should easily find results stating that for any deterministic time $t$, a Lévy process has almost surely no jumps at time $t$. The statement you're looking for is then an immediate corollary. – Martin Hairer Oct 6 '13 at 20:10
Thank you. I'm familiar with the fact that for a Levy process $X_t$ and some $s\in \mathbb{R}$ $\bigtriangleup X_{s}=0$ a.s. However, since the number of $s\in \mathbb{R}$ for which a jump occurred might not be measurable I don't see how it answers my question. – Ofer Oct 7 '13 at 11:32
use independence and fubini. – ofer zeitouni Oct 7 '13 at 12:03

1 Answer

Consider a finite time interval, let $J_i$ denote the set of (times of) jumps of the $i$'th process, and let $J_{i,n}$ denote the set of jumps of the $i$'th process of size $(\frac 1 n,\frac 1{n-1}]$. Note that $J_{i,n}$ is finite a.s. and therefore $J_{i,n}\cap J_{j,m}=\emptyset$ a.s. for all $n,m$ and $i\neq j$. Since there are countably many such intersections, it follows that $J_i\cap J_j = \emptyset$ a.s.

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@ mpr : I am not sure you are answering the question. Regards – The Bridge Oct 10 '13 at 8:02
The question, as I read it, was to show that independent Levy processes have no simultaneous jumps, which is to say that $J_i\cap J_j$ is empty. Maybe I am missing something. – mpr Oct 10 '13 at 11:25
$J_{i,n}$ is not finite, it could be countably infinite. You can see this from the Levy-Ito decomposition. – Bati Oct 10 '13 at 12:44
Countably infinite number of jumps of size greater than $1/n$ in a finite time interval? – mpr Oct 10 '13 at 12:57