  We propose an analytically tractable class of models for the dynamics of a
limit order book, described through a stochastic partial differential equation
(SPDE) with multiplicative noise for the order book centered at the mid-price,
along with stochastic dynamics for the mid-price which is consistent with the
order flow dynamics. We provide conditions under which the model admits a
finite dimensional realization driven by a (low-dimensional) Markov process,
leading to efficient estimation and computation methods. We study two examples
of parsimonious models in this class: a two-factor model and a model with
mean-reverting order book depth. For each model we analyze in detail the role
of different parameters, the dynamics of the price, order book depth, volume
and order imbalance, provide an intuitive financial interpretation of the
variables involved and show how the model reproduces statistical properties of
price changes, market depth and order flow in limit order markets.
