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Record statistics for biased random walks, with an application to financial data.
Gregor Wergen1, Miro Bogner, Joachim Krug
1Institut für Theoretische Physik, Universität zu Köln, Köln, Germany.
Summary
Record-breaking events in asymmetric random walks are analyzed. Unlike symmetric walks, asymmetric jump distributions alter record statistics, with results validated against stock market data.
Area of Science:
- Statistical Physics
- Stochastic Processes
Background:
- Record statistics in symmetric random walks are well-understood.
- Asymmetric jump distributions introduce complexities in record analysis.
Purpose of the Study:
- To compute the record rate for asymmetric random walks with Gaussian jumps.
- To analyze the dependence of record statistics on jump distribution asymmetry.
- To apply findings to real-world financial data.
Main Methods:
- Analytical computation of the record rate P(n)(c).
- Analysis of Gaussian jump distributions with standard deviation σ and drift c.
- Comparison of theoretical results with daily stock price data.
Main Results:
- Record rate P(n)(c) shows dependence on drift c and standard deviation σ.
- For small drift, corrections to P(n)(c) grow with arctan(√n).
- Asymptotic record rate approaches a constant for large drift, applicable to other continuous distributions.
Conclusions:
- Biased random walks quantitatively explain increased upper records in stock prices due to trends.
- Detrended stock data shows good agreement with symmetric random walk for upper records.
- A mechanism reducing lower records in detrended data requires further investigation.
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