{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,4,26]],"date-time":"2023-04-26T07:40:47Z","timestamp":1682494847643},"reference-count":15,"publisher":"Cambridge University Press (CUP)","issue":"03","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. Appl. Probab."],"published-print":{"date-parts":[[2012,9]]},"abstract":"<jats:p>Let (<jats:italic>X<\/jats:italic>\n               <jats:sub>\n                  <jats:italic>t<\/jats:italic>\n               <\/jats:sub>)<jats:sub>0 \u2264 <jats:italic>t<\/jats:italic> \u2264 <jats:italic>T<\/jats:italic>\n               <\/jats:sub>\nbe a one-dimensional stochastic process with independent and stationary increments, either in discrete or continuous time. In this paper we consider the problem of stopping the process (<jats:italic>X<\/jats:italic>\n               <jats:sub>\n                  <jats:italic>t<\/jats:italic>\n               <\/jats:sub>) \u2018as close as possible\u2019 to its eventual supremum <jats:italic>M<\/jats:italic>\n               <jats:sub>\n                  <jats:italic>T<\/jats:italic>\n               <\/jats:sub> := sup<jats:sub>0 \u2264 <jats:italic>t<\/jats:italic> \u2264 <jats:italic>T<\/jats:italic>\n               <\/jats:sub>\n               <jats:italic>X<\/jats:italic>\n               <jats:sub>\n                  <jats:italic>t<\/jats:italic>\n               <\/jats:sub>, when the reward for stopping at time \u03c4 \u2264 <jats:italic>T<\/jats:italic> is a nonincreasing convex function of <jats:italic>M<\/jats:italic>\n               <jats:sub>\n                  <jats:italic>T<\/jats:italic>\n               <\/jats:sub> - <jats:italic>X<\/jats:italic>\n               <jats:sub>\u03c4<\/jats:sub>. Under fairly general conditions on the process (<jats:italic>X<\/jats:italic>\n               <jats:sub>\n                  <jats:italic>t<\/jats:italic>\n               <\/jats:sub>), it is shown that the optimal stopping time \u03c4 takes a trivial form: it is either optimal to stop at time 0 or at time <jats:italic>T<\/jats:italic>. For the case of a random walk, the rule \u03c4 \u2261 <jats:italic>T<\/jats:italic> is optimal if the steps of the walk stochastically dominate their opposites, and the rule \u03c4 \u2261 0 is optimal if the reverse relationship holds. An analogous result is proved for L\u00e9vy processes with finite L\u00e9vy measure. The result is then extended to some processes with nonfinite L\u00e9vy measure, including stable processes, CGMY processes, and processes whose jump component is of finite variation.<\/jats:p>","DOI":"10.1017\/s0021900200009554","type":"journal-article","created":{"date-parts":[[2016,3,29]],"date-time":"2016-03-29T14:49:46Z","timestamp":1459262986000},"page":"806-820","source":"Crossref","is-referenced-by-count":0,"title":["Predicting the Supremum: Optimality of \u2018Stop at Once or Not at All\u2019"],"prefix":"10.1017","volume":"49","author":[{"given":"Pieter C.","family":"Allaart","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2016,2,4]]},"reference":[{"key":"S0021900200009554_ref12","doi-asserted-by":"publisher","DOI":"10.1080\/1045112031000118994"},{"key":"S0021900200009554_ref13","volume-title":"L\u00e9vy Processes and Infinitely Divisible Distributions","year":"1999"},{"key":"S0021900200009554_ref10","doi-asserted-by":"publisher","DOI":"10.1016\/0304-4149(88)90104-4"},{"key":"S0021900200009554_ref9","first-page":"41","volume":"45","year":"2000","journal-title":"Theory Prob. Appl."},{"key":"S0021900200009554_ref8","doi-asserted-by":"publisher","DOI":"10.1214\/08-AAP566"},{"key":"S0021900200009554_ref7","doi-asserted-by":"publisher","DOI":"10.1214\/009117906000000638"},{"key":"S0021900200009554_ref6","doi-asserted-by":"publisher","DOI":"10.1086\/338705"},{"key":"S0021900200009554_ref5","volume-title":"L\u00e9vy Processes","year":"1996"},{"key":"S0021900200009554_ref4","doi-asserted-by":"publisher","DOI":"10.1214\/10-AOP598"},{"key":"S0021900200009554_ref3","doi-asserted-by":"publisher","DOI":"10.1214\/07-AOP376"},{"key":"S0021900200009554_ref15","doi-asserted-by":"publisher","DOI":"10.1239\/jap\/1253279844"},{"key":"S0021900200009554_ref2","volume-title":"L\u00e9vy Processes and Stochastic Calculus","year":"2009"},{"key":"S0021900200009554_ref14","doi-asserted-by":"publisher","DOI":"10.1080\/14697680802563732"},{"key":"S0021900200009554_ref1","doi-asserted-by":"publisher","DOI":"10.1239\/jap\/1294170520"},{"key":"S0021900200009554_ref11","volume-title":"Inequalities: Theory of Majorization and Its Applications","year":"1979"}],"container-title":["Journal of Applied Probability"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0021900200009554","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,4,26]],"date-time":"2023-04-26T07:13:11Z","timestamp":1682493191000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0021900200009554\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,9]]},"references-count":15,"journal-issue":{"issue":"03","published-print":{"date-parts":[[2012,9]]}},"alternative-id":["S0021900200009554"],"URL":"https:\/\/doi.org\/10.1017\/s0021900200009554","relation":{},"ISSN":["0021-9002","1475-6072"],"issn-type":[{"value":"0021-9002","type":"print"},{"value":"1475-6072","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,9]]}}}