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 bidand ask price processes .
We consider a stochastic model for the dynamics of the two-sided limit order book (LOB). Our model is flexible enough to allow for a dependence of the price dynamics on volumes. 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 in distribution to a coupled SDE-SPDE system when the order arrival rates tend to infinity and the impact of an individual order arrival on the book as well as the tick size tends to zero. The SDE describes the bid/ask price dynamics while the SPDE describes the volume dynamics .
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Our model is flexible enough to allow for a dependence of the price dynamics on volumes.
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to a continuous-time limit
in distribution to a coupled SDE-SPDE system
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,
and
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tend
tends
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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 bidand ask price processes
SDE describes the bid/ask price dynamics while the SPDE describes the volume dynamics