This paper studies a one-sided limit order book (LOB) model, in which the order dynamics depend on both, the current best bid price and the current volume density function . For the joint dynamics of the best bid price  and the standing buy volume density we derive a weak law of large numbers, which states that the LOB model converges to a continuous-time limit when the size of an individual order as well as the tick size tend to zero and the order arrival rate tends to infinity. In the scaling limit the standing buy volume density follows a non-linear PDE coupled with a non-linear ODE that describes the best bid  price.
This paper studies a limit order book (LOB) model, in which the order dynamics depend on both, the current best available prices and the current volume density functions . For the joint dynamics of the best bid price , the best ask price, and the standing volume densities on both sides of the LOB we derive a weak law of large numbers, which states that the LOB model converges to a continuous-time limit when the size of an individual order as well as the tick size tend to zero and the order arrival rate tends to infinity. In the scaling limit the two volume densities follow each a non-linear PDE coupled with two non-linear ODEs that describe the best bid and ask price.
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one-sided
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bid price
available prices
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function
functions
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, the best ask price,
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buy volume density
volume densities on both sides of the LOB
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standing buy volume density follows
two volume densities follow each
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a
two
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ODE that describes
ODEs that describe
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and ask