Diffusive Limit of Hawkes Driven Order Book Dynamics With Liquidity Migration

This paper develops a theoretical mesoscopic model of the limit order book driven by multivariate Hawkes processes, designed to capture temporal self-excitation and the spatial propagation of order flow across price levels. In contrast to classical zero-intelligence or Poisson based queueing models, the proposed framework introduces mathematically defined migration events between neighbouring price levels, whose intensities are themselves governed by the underlying Hawkes structure. This provides a principled stochastic mechanism for modeling interactions between order arrivals, cancellations, and liquidity movement across adjacent queues. Starting from a microscopic specification of Hawkes driven order flow, we derive a diffusion approximation which yields a reflected mesoscopic stochastic differential equation (SDE) system for queue volumes. The limiting generator is obtained through a Taylor expansion of the microscopic generator, demonstrating how temporal excitation together with spatial migration determine the drift and diffusion structure of the limit order book in the mesoscopic regime. The resulting model extends existing diffusion limits by incorporating correlated excitations and price level to price level liquidity movement within a unified Hawkes based formulation. By establishing this diffusive limit, the paper provides a mathematically consistent bridge between high frequency event based models and macroscopic stochastic descriptions of market microstructure. The work is entirely theoretical and lays a foundation for future analytical and numerical developments without relying on empirical calibration.