  We consider a process $X_t$, which is observed on a finite time interval
$[0,T]$, at discrete times $0,\Delta_n,2\Delta_n,\ldots.$ This process is an
It\^{o} semimartingale with stochastic volatility $\sigma_t^2$. Assuming that
$X$ has jumps on $[0,T]$, we derive tests to decide whether the volatility
process has jumps occurring simultaneously with the jumps of $X_t$. There are
two different families of tests for the two possible null hypotheses (common
jumps or disjoint jumps). They have a prescribed asymptotic level as the mesh
$\Delta_n$ goes to $0$. We show on some simulations that these tests perform
reasonably well even in the finite sample case, and we also put them in use on
S&P 500 index data.
