标题： 记录有偏随机游走的统计信息，应用于金融数据 显示英文标题

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

摘要： 我们考虑在跳跃分布不对称的随机漫步中记录突破事件的发生。 对称随机漫步中的记录统计之前由Majumdar和Ziff分析过，并且已经被很好地理解。 与对称跳跃分布的情况不同，在不对称情况下，记录的统计依赖于跳跃分布的选择。 我们计算记录率$P_n(c)$，定义为第$n$个值大于所有先前值的概率，对于标准差为$\sigma$的高斯跳跃分布，该分布被一个常数漂移$c$所偏移。 对于小漂移，在$c/\sigma \ll n^{-1/2}$的意义上，对$P_n(c)$的修正与 arctan$(\sqrt{n})$成正比，并在值$\frac{c}{\sqrt{2} \sigma}$处达到饱和。 对于大的$n$，记录率趋近于一个常数，该常数大约由$1-(\sigma/\sqrt{2\pi}c)\textrm{exp}(-c^2/2\sigma^2)$给出，对于$c/\sigma \gg 1$而言。 这些渐近结果适用于其他具有有限方差的连续跳跃分布。 作为应用，我们将分析结果与标准普尔500指数的366个每日股价的记录统计进行了比较。 有偏随机游走定性地解释了由于股价整体趋势导致的上界记录数量的增加，去趋势后上界记录的数量与对称随机游走的结果良好一致。 然而，去趋势数据中的下界记录数量显著减少，这一机制仍有待确定。 显示英文摘要

摘要： 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.