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Second order approximations for limit order books

Finance and Stochastics volume 22, pages 827–877 (2018)Cite this article

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.

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Notes

  1. In [12], the claim was proved for \(p^{(n)}\equiv p\) not depending on . However, the proof in [12] can easily be extended to more general \(p^{(n)}\) as long as \(p^{(n)}\) converges uniformly to some \(p\).

  2. By ‘weak’ solution, we mean ‘weak’ in the PDE sense, i.e., we consider \(Z^{u}\) as a distribution-valued process, being an element of \(H^{-3}\). The exact solution concept is explicitly defined in Theorem 6.9 below.

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Authors and Affiliations

  1. Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099, Berlin, Germany

    Ulrich Horst & Dörte Kreher

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  1. Ulrich Horst
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  2. Dörte Kreher
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Correspondence to Ulrich Horst.

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Financial support through the CRC TRR 190 is gratefully acknowledged.

Appendix A: A technical estimate for \(L^{2}\)-martingales

Appendix A: A technical estimate for \(L^{2}\)-martingales

Lemma A.1

([21, Theorem 6.1])

There exists a constant \(C > 0\) such that for all martingale differences \((X_{i})\) with values in , we have

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Horst, U., Kreher, D. Second order approximations for limit order books. Finance Stoch 22, 827–877 (2018). https://doi.org/10.1007/s00780-018-0373-7

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