Modelling bid and ask prices using constrained Hawkes processes: Ergodicity and scaling limit

We introduce a multivariate point process describing the dynamics of the Bid and Ask price of a financial asset. The point process is similar to a Hawkes process, with additional constraints on its intensity corresponding to the natural ordering of the best Bid and Ask prices. 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 including the constraint variable. Natural, explicit conditions on the parameters are established that ensure the ergodicity of the chain. Moreover, scaling limits are derived for the integrated point process.