Hydrodynamic limit of order book dynamics

Motivated by optimal trade execution, this paper studies the temporal evolution of the shape of a limit order book over a time horizon that is large compared with the length of time between order book events, with the aim of approximating the transient distribution of the shape. Relying on the stochastic order book model in Cont et al. (2010), we show that when the tick size approaches zero, a pair of measure-valued processes representing the "sell-side shape" and "buy-side shape" of an order book converges to a pair of deterministic measure-valued processes in a certain sense. Moreover, we describe the density profile of the limiting processes through ordinary differential equations which can be solved explicitly. We also perform experiments to test our limiting model against data. The empirical results suggest that the approximation is effective.