A Scaling Limit for Limit Order Books Driven by Hawkes Processes

Abstract

In this paper we derive a scaling limit for an infinite-dimensional limit order book model driven by Hawkes random measures. The dynamics of the incoming order flow is allowed to depend on the current market price as well as on a volume indicator. With our choice of scaling the dynamics converges to a coupled SDE-ODE system where limiting best bid and ask price processes follows a diffusion dynamics, the limiting volume density functions follows an ODE in a Hilbert space, and the limiting order arrival and cancellation intensities follow a Volterra--Fredholm integral equation.