Second order approximations for limit order books

Ulrich Horst () and Dörte Kreher ()
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Ulrich Horst: Humboldt-Universität zu Berlin
Dörte Kreher: Humboldt-Universität zu Berlin

Finance and Stochastics, 2018, vol. 22, issue 4, No 3, 827-877

Abstract: Abstract In this paper, we derive a second order approximation for an infinite-dimensional limit order book model, in which the dynamics of the incoming order flow is allowed to depend on the current market price as well as on a volume indicator (e.g. the volume standing at the top of the book). We study the fluctuations of the price and volume process relative to their first order approximation given in ODE–PDE form under two different scaling regimes. In the first case, we suppose that price changes are really rare, yielding a constant first order approximation for the price. This leads to a measure-valued SDE driven by an infinite-dimensional Brownian motion in the second order approximation of the volume process. In the second case, we use a slower rescaling rate, which leads to a non-degenerate first order approximation and gives a PDE with random coefficients in the second order approximation for the volume process. Our results can be used to derive confidence intervals for models of optimal portfolio liquidation under market impact.

Keywords: Functional central limit theorem; Second order approximation; High-frequency limit; Limit order book; 60F17; 91G80 (search for similar items in EconPapers)
JEL-codes: G10 G12 (search for similar items in EconPapers)
Date: 2018
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Citations: View citations in EconPapers (5)

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DOI: 10.1007/s00780-018-0373-7

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