预测随机游动最大值的一般“bang-bang”原理
A general "bang-bang" principle for predicting the maximum of a random walk
摘要：
设$(B_t)_{0\leq t\leq T}$是Bernoulli随机游动或具有漂移的布朗运动，并设$M_t:=\max\{B_s: 0\leq s\leq t\}$，$0\leq t\leq T$。本文解决了一般最优预测问题\sup_{0\leqτ\leq T} \sE [f（M_T-B_τ）]，其中上确界是在适合于$(B_t)$的自然过滤的所有停时$τ$上，$f$是一个非增凸函数.最优停止时间$τ^*$被证明是“bang-bang”型的：如果底层过程$(B_t)$的漂移是负的，则为$τ^*\equiv 0$，如果漂移是正的，则为$τ^*\equiv T$。这一结果推广了S. Yam，S. Yung和W. Zhou [{ \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），并为金融学中的格言提供了额外的数学证明，即应该立即出售坏股票，但尽可能长时间地保留好股票。
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\leqτ\leq T}\sE[f(M_T-B_τ)], where the supremum is over all stopping times $τ$ adapted to the natural filtration of $(B_t)$, and $f$ is a nonincreasing convex function. The optimal stopping time $τ^*$ is shown to be of "bang-bang" type: $τ^*\equiv 0$ if the drift of the underlying process $(B_t)$ is negative, and $τ^*\equiv T$ 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.
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。
