A General ‘Bang-Bang’ Principle for Predicting the Maximum of a Random Walk

Content Provider	Scilit
Author	Allaart, Pieter
Copyright Year	2010
Description	Let (B$ _{ t })_{0≤t≤T }$ be either a Bernoulli random walk or a Brownian motion with drift, and let M$ _{ t }$ := max{B$ _{s}$: 0 ≤ s ≤ t}, 0 ≤ t ≤ T. In this paper we solve the general optimal prediction problem $sup_{0≤τ≤T }$E[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: $τ^{*}$ ≡ 0 if the drift of the underlying process (B$ _{ t }$) is negative and $τ^{*}$ ≡ T if the drift is positive. This result generalizes recent findings of Toit and Peskir (2009) and Yam, Yung and Zhou (2009), and provides additional mathematical justification for the dictum in finance that one should sell bad stocks immediately, but keep good stocks as long as possible.
Related Links	https://www.cambridge.org/core/services/aop-cambridge-core/content/view/F6AF2F86D81E14B241BFEF36E9CD395A/S0021900200007373a.pdf/div-class-title-a-general-bang-bang-principle-for-predicting-the-maximum-of-a-random-walk-div.pdf
Ending Page	1083
Page Count	12
Starting Page	1072
ISSN	00219002
e-ISSN	14756072
DOI	10.1017/s0021900200007373
Journal	Journal of applied probability
Issue Number	04
Volume Number	47
Language	English
Publisher	Cambridge University Press (CUP)
Publisher Date	2010-03-01
Access Restriction	Open
Subject Keyword	Journal of applied probability Operations Research and Management Science Bernoulli Random Walk Brownian Motion Optimal Prediction Ultimate Maximum Stopping Time Convex Function
Content Type	Text
Resource Type	Article
Subject	Statistics and Probability Statistics, Probability and Uncertainty