A non-linear model of limit order book dynamics
  This paper focuses on some simple models of limit order book dynamics which simulate market trading mechanisms. We start with a discrete time/space Markov process and then perform a re-scaling procedure leading to a deterministic dynamical system controlled by non-linear ODEs. This allows us to introduce approximants for the equilibrium distribution of the process represented by fixed points of deterministic dynamics. 
N Vvedenskaya, Y Suhov, V Belitsky
math.PR