  In this paper we derive a second order approximation for an infinite
dimensional limit order book model, in which the dynamics of the incoming order
flow is allowed to depend on the current market price as well as on a volume
indicator (e.g.~the volume standing at the top of the book). We study the
fluctuations of the price and volume process relative to their first order
approximation given in ODE-PDE form under two different scaling regimes. In the
first case we suppose that price changes are really rare, yielding a constant
first order approximation for the price. This leads to a measure-valued SDE
driven by an infinite dimensional Brownian motion in the second order
approximation of the volume process. In the second case we use a slower
rescaling rate, which leads to a non-degenerate first order approximation and
gives a PDE with random coefficients in the second order approximation for the
volume process. Our results can be used to derive confidence intervals for
models of optimal portfolio liquidation under market impact.
