摘要翻译：
设$(B_t)_{0\leqt\leqt}$为Bernoulli随机游动或带漂移的布朗运动，且设$m_t:=\max\{b_s:0\leqs\leqt\}$,$0\leqt\leqt$。本文解决了一般的最优预测问题SUP_{0\leq\tau\leqt}\se[f(M_t-b_tau)],其中上确界在所有停止时间$\tau$上，适用于$(B_t)$的自然过滤，且$f$是一个非增凸函数。最佳停止时间$\tau^*$显示为“bang-bang”类型:$\tau^*\equiv0$，如果基础进程$(B_t)$的漂移为负，且$\tau^*\equivt$为漂移为正。这一结果概括了S.Yam、S.Yung和W.Zhou的最新发现[{{\em J.appl.probab.}{\bf 46}（2009）,651-668]和J.Du Toit和G.Peskir[{{em Ann.appl.probab.}{bf 19}（2009）,983-1014],并为金融学中的一个格言提供了额外的数学依据，即一个人应该立即卖出坏股票，但要尽可能长时间地保留好股票。
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英文标题：
《A general "bang-bang" principle for predicting the maximum of a random
  walk》
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作者：
Pieter C. Allaart
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最新提交年份：
2009
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分类信息：

一级分类：Mathematics        数学
二级分类：Probability        概率
分类描述：Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory
概率论与随机过程的理论与应用：例如中心极限定理，大偏差，随机微分方程，统计力学模型，排队论
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一级分类：Quantitative Finance        数量金融学
二级分类：Portfolio Management        项目组合管理
分类描述：Security selection and optimization, capital allocation, investment strategies and performance measurement
证券选择与优化、资本配置、投资策略与绩效评价
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一级分类：Quantitative Finance        数量金融学
二级分类：Statistical Finance        统计金融
分类描述：Statistical, econometric and econophysics analyses with applications to financial markets and economic data
统计、计量经济学和经济物理学分析及其在金融市场和经济数据中的应用
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英文摘要：
  Let $(B_t)_{0\leq t\leq T}$ be either a Bernoulli random walk or a Brownian motion with drift, and let $M_t:=\max\{B_s: 0\leq s\leq t\}$, $0\leq t\leq T$. This paper solves the general optimal prediction problem \sup_{0\leq\tau\leq T}\sE[f(M_T-B_\tau)], where the supremum is over all stopping times $\tau$ adapted to the natural filtration of $(B_t)$, and $f$ is a nonincreasing convex function. The optimal stopping time $\tau^*$ is shown to be of "bang-bang" type: $\tau^*\equiv 0$ if the drift of the underlying process $(B_t)$ is negative, and $\tau^*\equiv T$ is the drift is positive. This result generalizes recent findings by S. Yam, S. Yung and W. Zhou [{\em J. Appl. Probab.} {\bf 46} (2009), 651--668] and J. Du Toit and G. Peskir [{\em Ann. Appl. Probab.} {\bf 19} (2009), 983--1014], and provides additional mathematical justification for the dictum in finance that one should sell bad stocks immediately, but keep good ones as long as possible.
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PDF链接：
https://arxiv.org/pdf/0910.0545