Jump diffusion approximation for the price dynamics of a fully state dependent limit order book model

Abstract

We study a one-sided discrete-time limit order book model, in which the order dynamics depend on the current best available price and the current volume density function, simultaneously. In order to take extreme price movements into account, we include price changes to our model which do not scale with the tick size. We derive a functional convergence result, which states that the limit order book model converges to a continuous-time limit when the size of an individual order as well as the tick size tend to zero and the order arrival rate tends to infinity. In the high frequency limit the order book dynamics are described by a one-dimensional jump diffusion characterizing the price process coupled with an infinite-dimensional fluid process characterizing the standing volumes.