  We introduce a multivariate Hawkes process with constraints on its
conditional density. It is a multivariate point process with conditional
intensity similar to that of a multivariate Hawkes process but certain events
are forbidden with respect to boundary conditions on a multidimensional
constraint variable, whose evolution is driven by the point process. We study
this process in the special case where the fertility function is exponential so
that the process is entirely described by an underlying Markov chain, which
includes the constraint variable. Some conditions on the parameters are
established to ensure the ergodicity of the chain. Moreover, scaling limits are
derived for the integrated point process. This study is primarily motivated by
the stochastic modelling of a limit order book for high frequency financial
data analysis.
