A law of large numbers for limit order books

We define a stochastic model of a two-sided limit order book in terms of its key quantities $\mathit{\text{best}}\mathit{\text{bid}}\mathit{\text{[ask]}}\mathit{\text{price}}$ and the \textit{standing buy [sell] volume density}. For a simple scaling of the discreteness parameters, that keeps the expected volume rate over the considered price interval invariant, we prove a limit theorem. The limit theorem states that, given regularity conditions on the random order flow, the key quantities converge in probability to a tractable continuous limiting model. In the limit model the buy and sell volume densities are given as the unique solution to first-order linear hyperbolic PDEs, specified by the expected order flow parameters.