  This paper studies a limit order book (LOB) model, in which the order
dynamics depend on both, the current best available prices and the current
volume density functions. For the joint dynamics of the best bid price, the
best ask price, and the standing volume densities on both sides of the LOB we
derive a weak law of large numbers, which states that the LOB 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
scaling limit the two volume densities follow each a non-linear PDE coupled
with two non-linear ODEs that describe the best bid and ask price.
