摘要翻译：
假定股票价格$Z=(Z_t)_{0\leqt\leqt}$服从具有漂移$\mu\in\mathbb{R}$和波动率$\sigma>0$的几何布朗运动，并设$M_T=\max_{0\leqs\leqt}z_s$为$T\in[0,T]$,我们考虑了最优预测问题\[v_1=\inf_{0\leq\tau\leqt}\mathsf{E}\biggl(\frac{M_T}{Z_{\tau}}\biggr)\quadand\quad v_2=\sup_{0\leq\tau\leqt{E}\biggl(\frac{Z_{\tau}}{M_T}\biggr),\]其中的下确界和上确界将接管$z$的所有停止时间$\tau$。我们证明了以下策略在第一个问题中是最优的：如果$\mu\leq0$立即停止；如果$\mu\in(0,\sigma^2)$在$m_t/z_t$达到指定的时间函数时立即停止；如果$\mu\geq\sigma^2$等到最后一次$t$。通过对比，我们证明了以下策略在第二个问题中是最优的：如果$\mu\leq\sigma^2/2$立即停止，如果$\mu>\sigma^2/2$等到最后一次$t$。这两种解决方案都支持并强化了人们普遍持有的金融观点，即“一个人应该卖出坏股票，保留好股票”。证明方法利用了抛物型自由边界问题和局部时空演算技术。由此产生的不平等就其本身而言是不寻常和有趣的，因为它们涉及未来，因此具有预测因素。
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英文标题：
《Selling a stock at the ultimate maximum》
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作者：
Jacques du Toit, Goran Peskir
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最新提交年份：
2009
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分类信息：

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：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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英文摘要：
  Assuming that the stock price $Z=(Z_t)_{0\leq t\leq T}$ follows a geometric Brownian motion with drift $\mu\in\mathbb{R}$ and volatility $\sigma>0$, and letting $M_t=\max_{0\leq s\leq t}Z_s$ for $t\in[0,T]$, we consider the optimal prediction problems \[V_1=\inf_{0\leq\tau\leq T}\mathsf{E}\biggl(\frac{M_T}{Z_{\tau}}\biggr)\quadand\quad V_2=\sup_{0\leq\tau\leq T}\mathsf{E}\biggl(\frac{Z_{\tau}}{M_T}\biggr),\] where the infimum and supremum are taken over all stopping times $\tau$ of $Z$. We show that the following strategy is optimal in the first problem: if $\mu\leq0$ stop immediately; if $\mu\in (0,\sigma^2)$ stop as soon as $M_t/Z_t$ hits a specified function of time; and if $\mu\geq\sigma^2$ wait until the final time $T$. By contrast we show that the following strategy is optimal in the second problem: if $\mu\leq\sigma^2/2$ stop immediately, and if $\mu>\sigma^2/2$ wait until the final time $T$. Both solutions support and reinforce the widely held financial view that ``one should sell bad stocks and keep good ones.'' The method of proof makes use of parabolic free-boundary problems and local time--space calculus techniques. The resulting inequalities are unusual and interesting in their own right as they involve the future and as such have a predictive element.
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PDF链接：
https://arxiv.org/pdf/0908.1014