Queue-Reactive Dynamics on the Full Order-Book Profile

Queue-reactive models of the limit order book make order arrival and cancellation rates functions of the current queue size, and reproduce the mean reversion of depth that constant-rate models miss. They are, however, formulated at the best bid and best ask only: they carry no depth, and therefore no coupling between adjacent price levels. The reflected-SPDE family carries both, but specifies rates that depend on distance from the mid alone and are linear in volume. We give a model that holds both properties at once — rates depending jointly on position and on local queue size, adjacent levels coupled by a discrete Laplacian, volumes kept non-negative without a reflection measure, and the mid-price moving only when a best queue is depleted. We prove the model contains each parent family as a parameter restriction, and that a specification whose rates do not depend on the state has a stationary law of a single shape: its coefficient of variation is pinned at unity whatever its parameters, so the stationary mean and variance of depth cannot be matched independently. This is a structural account of the over-estimation of profile volatility reported for scaling-limit reductions in which the state dependence is lost. We further show that the sine eigenbasis usually assumed for the linearised generator is a property of uniform cancellation alone: once cancellation varies with position, the modes localise. Every claim is verified by simulation, and an acceptance test against order-by-order data is stated in advance of any such data being held.