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Record statistics for biased random walks, with an application to financial data

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  • Gregor Wergen
  • Miro Bogner
  • Joachim Krug

Abstract

We consider the occurrence of record-breaking events in random walks with asymmetric jump distributions. The statistics of records in symmetric random walks was previously analyzed by Majumdar and Ziff and is well understood. Unlike the case of symmetric jump distributions, in the asymmetric case the statistics of records depends on the choice of the jump distribution. We compute the record rate $P_n(c)$, defined as the probability for the $n$th value to be larger than all previous values, for a Gaussian jump distribution with standard deviation $\sigma$ that is shifted by a constant drift $c$. For small drift, in the sense of $c/\sigma \ll n^{-1/2}$, the correction to $P_n(c)$ grows proportional to arctan$(\sqrt{n})$ and saturates at the value $\frac{c}{\sqrt{2} \sigma}$. For large $n$ the record rate approaches a constant, which is approximately given by $1-(\sigma/\sqrt{2\pi}c)\textrm{exp}(-c^2/2\sigma^2)$ for $c/\sigma \gg 1$. These asymptotic results carry over to other continuous jump distributions with finite variance. As an application, we compare our analytical results to the record statistics of 366 daily stock prices from the Standard & Poors 500 index. The biased random walk accounts quantitatively for the increase in the number of upper records due to the overall trend in the stock prices, and after detrending the number of upper records is in good agreement with the symmetric random walk. However the number of lower records in the detrended data is significantly reduced by a mechanism that remains to be identified.

Suggested Citation

  • Gregor Wergen & Miro Bogner & Joachim Krug, 2011. "Record statistics for biased random walks, with an application to financial data," Papers 1103.0893, arXiv.org.
  • Handle: RePEc:arx:papers:1103.0893
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    Cited by:

    1. Claude Godreche & Satya N. Majumdar & Gregory Schehr, 2017. "Record statistics of a strongly correlated time series: random walks and L\'evy flights," Papers 1702.00586, arXiv.org.
    2. Min, Seungsik & Shin, Ki-Hong & Baek, Woonhak & Kim, Kyungsik & You, Cheol-Hwan & Lee, Dong-In & Yum, Seong Soo & Kim, Wonheung & Chang, Ki-Ho, 2020. "Dynamical behavior of combined detrended cross-correlation analysis methods in random walks and Lévy flights," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 539(C).
    3. Wergen, Gregor, 2014. "Modeling record-breaking stock prices," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 396(C), pages 114-133.
    4. Srivastava, Shashi C.L. & Lakshminarayan, Arul, 2015. "Records in the classical and quantum standard map," Chaos, Solitons & Fractals, Elsevier, vol. 74(C), pages 67-78.
    5. Claude Godreche & Satya N. Majumdar & Gregory Schehr, 2015. "Record statistics for random walk bridges," Papers 1505.06053, arXiv.org, revised Jan 2016.
    6. Gregory Schehr & Satya N. Majumdar, 2013. "Exact record and order statistics of random walks via first-passage ideas," Papers 1305.0639, arXiv.org.

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