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

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 [1] 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 nth value to be larger than all previous values, for a Gaussian jump distribution with standard deviation σ that is shifted by a constant drift c. For small drift, in the sense of c/σ ≪ n −1/2 , the correction to P n (c) grows proportional to arctan( √ n) and saturates at the value. For large n the record rate approaches a constant, which is approximately given by 1 − σ/ √ 2πc exp −c 2 /2σ 2 for c/σ ≫ 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.

Records in fractal stochastic processes

“…However, many time series in real situations are correlated. In this respect, records in symmetric random walks has been well studied, recently [39], which triggers a series of new research studies, e.g., record statistics of biased random walks and Levy flights [21,22], multiple independent * manshour@pgu.ac.ir random walks [21], and continuous-time random walks [40]. The main aim of such studies is to understand and model the statistics of records in observational data by comparing with various kinds of stochastic processes.…”

“…Recently, the extreme event studies have been developed using various approaches, such as the density of states [3,4], firstpassage and return-time statistics [6][7][8], persistence [9], interoccurrence time statistics [10][11][12] and record statistics [13][14][15]. Among them, there has been considerable interest in investigating the record statistics, which has found many fruitful applications in diverse complex systems, such as spin glasses [16,17], adaptive processes [18], domain wall dynamics [19], avalanche dynamics [20], stock prices [21,22], global warming [23,24], growing network [25], high-temperature superconductors [26], the ant movements dynamics [27], flood dynamics [28], sport statistics [29,30], earthquakes [31,32], and evolutionary biology [33,34].…”

Record statistics of financial time series and geometric random walks

“…Significantly, it is independent of the value of N . In contrast to quantities like the mean number of records which depend on N [7,12,15], the distribution of record ages is characteristic statistical property of record breaking events independent of the length of data. Further, as N increases, the range over which the power law is valid also increases implying that the tail behavior is a finite size effect.…”

“…Record age distribution will provide insights into how long a record can be expected to survive and is useful in hazard estimation problems. Though the mean record age has been analytically determined for random walk problems in earlier works [7,15], there have been no results for the distribution of record age.…”

“…These results have further been generalized to the case of random walk with a constant drift with application to stock market data [15] and also to multiple random walkers [11].…”