偏置扩散在首次退出时间前所扫面积：鞅方法与赌博机会
On the area swept by a biased diffusion till its first-exit time: Martingale approach and gambling opportunities
AI总结：
本文用鞅理论导出一维偏置扩散首次退出时间、位置及含记忆泛函的精确统计量，并据此求出任意非对称区间内平均扫面积与时间—位置互相关，还揭示其构造平均净赢赌博策略的应用。
摘要：
利用鞅理论，我们仅用寥寥数行便计算出与一维偏置扩散相关的各种首次退出时间统计量的精确解析表达式。例子包括从区间首次退出时间的分布、首次退出位置的各阶矩，以及涉及记忆和时间积分的位置泛函。作为关键例子，我们解析地计算了偏置扩散在逃出一个可能非对称且具有任意长度的区间之前所扫面积的均值。该平均面积使我们能够推导迄今尚未研究的首次退出时间与首次退出位置之间的互相关函数；该函数仅在对称区间的退出问题中为零。作为收尾，我们探讨了所得结果与赌博之间的联系，表明对一个亏损游戏的时间积分值下注，可以设计出一种带来平均净赢的策略。
英文摘要：
Using martingale theory, we compute, in very few lines, exact analytical expressions for various first-exit-time statistics associated with one-dimensional biased diffusion. Examples include the distribution for the first-exit time from an interval, moments for the first-exit site, and functionals of the position, which involve memory and time integration. As a key example, we compute analytically the mean area swept by a biased diffusion until it escapes an interval that may be asymmetric and have arbitrary length. The mean area allows us to derive the hitherto unexplored cross-correlation function between the first-exit time and the first-exit site, which vanishes only for exit problems from symmetric intervals. As a colophon, we explore connections of our results with gambling, showing that betting on the time-integrated value of a losing game it is possible to design a strategy that leads to a net average win.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
