A buffer Hawkes process for limit order books

We introduce a Markovian single point process model, with random intensity regulated through a buffer mechanism and a self-exciting effect controlling the arrival stream to the buffer. The model applies the principle of the Hawkes process in which point process jumps generate a shot-noise intensity field. Unlike the Hawkes case, the intensity field is fed into a separate buffer, the size of which is the driving intensity of new jumps. In this manner, the intensity loop portrays mutual-excitation of point process events and buffer size dynamics. This scenario is directly applicable to the market evolution of limit order books, with buffer size being the current number of limit orders and the jumps representing the execution of market orders. We give a branching process representation of the point process and prove that the scaling limit is Brownian motion with explicit volatility.