  Quadratic Hawkes (QHawkes) processes have proved effective at reproducing the
statistics of price changes, capturing many of the stylised facts of financial
markets. Motivated by the recently reported strong occurrence of endogenous
co-jumps (simultaneous price jumps of several assets) we extend QHawkes to a
multivariate framework (MQHawkes), that is considering several financial assets
and their interactions. Assuming that quadratic kernels write as the sum of a
time-diagonal component and a rank one (trend) contribution, we investigate
endogeneity ratios and the resulting stationarity conditions. We then derive
the so-called Yule-Walker equations relating covariances and feedback kernels,
which are essential to calibrate the MQHawkes process on empirical data.
Finally, we investigate the volatility distribution of the process and find
that, as in the univariate case, it exhibits power-law behavior, with an
exponent that can be exactly computed in some limiting cases.
