WIAS Preprint No. 1954, (2014)

A functional limit theorem for limit order books


  • Bayer, Christian
    ORCID: 0000-0002-9116-0039
  • Horst, Ulrich
  • Qiu, Jinniao

2010 Mathematics Subject Classification

  • 60B11 90B22 91B70


  • Limit order book, Scaling limit, Averaging principle, Queuing theory




We consider a stochastic model for the dynamics of the two-sided limit order book (LOB). For the joint dynamics of best bid and ask prices and the standing buy and sell volume densities, we derive a functional limit theorem, which states that our LOB model converges to a continuous-time limit when the order arrival rates tend to infinity, the impact of an individual order arrival on the book as well as the tick size tend to zero. The limits of the standing buy and sell volume densities are described by two linear stochastic partial differential equations, which are coupled with a two-dimensional reflected Brownian motion that is the limit of the best bid and ask price processes.

Appeared in

  • Ann. Appl. Probab., 27 (2017) pp. 2753-2806.

Download Documents