A Limit Theorem for Continuum Limit Order Book Dynamics

Summary

The document defines a stochastic model of a two-sided limit order book using best bid and ask prices and the standing buy and sell volume densities. It then considers a scaling of the model's discreteness parameters that preserves the expected volume rate over the price interval under study.

Under regularity conditions on random order flow, the authors prove that these quantities converge in probability to a continuous limiting model. In that limit, the buy and sell volume densities are characterized as the unique solutions of first-order linear hyperbolic partial differential equations, with coefficients or specifications determined by expected order flow parameters. The result provides a tractable continuum description derived from a discrete stochastic book. The excerpt does not state the precise regularity conditions or discuss empirical fit, so the theorem's applicability depends on assumptions not detailed here.

Key ideas

• The stochastic book is described by best bid and ask prices and standing volume densities on each side.
• A scaling that preserves expected volume rate leads to a continuum limit.
• Under regularity conditions, key book quantities converge in probability to the limiting model.
• The limiting buy and sell volume densities solve unique first-order linear hyperbolic PDEs determined by expected order flow.

Tags

# A law of large numbers for limit order books


# A law of large numbers for limit order books









We define a stochastic model of a two-sided limit order book in terms of its key quantities \textit{best bid [ask] price} and the \textit{standing buy [sell] volume density}. For a simple scaling of the discreteness parameters, that keeps the expected volume rate over the considered price interval invariant, we prove a limit theorem. The limit theorem states that, given regularity conditions on the random order flow, the key quantities converge in probability to a tractable continuous limiting model. In the limit model the buy and sell volume densities are given as the unique solution to first-order linear hyperbolic PDEs, specified by the expected order flow parameters.

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