Diffusion Limit of Poisson Limit-Order Book Models

Trading of financial instruments has largely moved away from floor trading and onto electronic exchanges. Orders to buy and sell are queued at these exchanges in a {\em limit-order book}. While a full analysis of the dynamics of a limit-order book requires an understanding of strategic play among multiple agents, and is thus extremely complex, so-called {\em zero-intelligence Poisson models} have been shown to capture many of the statistical features of limit-order book evolution. These models can be addressed by traditional queueing theory techniques, including Laplace transform analysis. In this article, we demonstrate in a simple setting that another queueing theory technique, approximating the Poisson model by a diffusion model identified as the limit of a sequence of scaled Poisson models, can also be implemented. We identify the diffusion limit, find an embedded semi-Markov model in the limit, and determine the statistics of the embedded semi-Markov model. Along the way, we introduce and study a new type of process, a generalization of skew Brownian motion that we call {\em two-speed Brownian motion}.