{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,9,4]],"date-time":"2023-09-04T06:50:41Z","timestamp":1693810241917},"reference-count":19,"publisher":"Association for Computing Machinery (ACM)","issue":"3","license":[{"start":{"date-parts":[[2018,7,13]],"date-time":"2018-07-13T00:00:00Z","timestamp":1531440000000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"funder":[{"DOI":"10.13039\/501100003246","name":"Netherlands Organisation for Scientific Research","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100003246","id-type":"DOI","asserted-by":"crossref"}]},{"name":"NWO Gravitation Programme NETWORKS","award":["024.002.003"]}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Trans. Model. Comput. Simul."],"published-print":{"date-parts":[[2018,7,31]]},"abstract":"We consider the bias arising from time discretization when estimating the threshold crossing probabilityw<\/jats:italic>(b<\/jats:italic>) := P(supt<\/jats:italic>\u03f5 [0,1]<\/jats:sub>B<\/jats:italic>t<\/jats:italic><\/jats:sub>>b<\/jats:italic>), with (B<\/jats:italic>t<\/jats:italic><\/jats:sub>)t<\/jats:italic>\u2208 [0,1]<\/jats:sub>a standard Brownian Motion. We prove that if the discretization is equidistant, then to reach a given target value of the relative bias, the number of grid points has to grow quadratically inb<\/jats:italic>, asb<\/jats:italic>grows. When considering non-equidistant discretizations (with threshold-dependent grid points), we can substantially improve on this: we show that for such grids the required number of grid points is independent ofb<\/jats:italic>, and in addition we point out how they can be used to construct a strongly efficient algorithm for the estimation ofw<\/jats:italic>(b<\/jats:italic>). Finally, we show how to apply the resulting algorithm for a broad class of stochastic processes; it is empirically shown that the threshold-dependent grid significantly outperforms its equidistant counterpart.<\/jats:p>","DOI":"10.1145\/3177775","type":"journal-article","created":{"date-parts":[[2018,7,13]],"date-time":"2018-07-13T16:08:17Z","timestamp":1531498097000},"page":"1-25","update-policy":"http:\/\/dx.doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":1,"title":["Controlling the Time Discretization Bias for the Supremum of Brownian Motion"],"prefix":"10.1145","volume":"28","author":[{"ORCID":"http:\/\/orcid.org\/0000-0002-0669-4248","authenticated-orcid":false,"given":"Krzysztof","family":"Bisewski","sequence":"first","affiliation":[{"name":"CWI Amsterdam, Amsterdam, Netherlands"}]},{"given":"Daan","family":"Crommelin","sequence":"additional","affiliation":[{"name":"CWI Amsterdam, KdV Institute for Mathematics, University of Amsterdam, Amsterdam, Netherlands"}]},{"given":"Michel","family":"Mandjes","sequence":"additional","affiliation":[{"name":"KdV Institute for Mathematics, University of Amsterdam, Amsterdam, Netherlands"}]}],"member":"320","published-online":{"date-parts":[[2018,7,13]]},"reference":[{"key":"e_1_2_2_1_1","doi-asserted-by":"crossref","volume-title":"An Introduction to Continuity, Extrema, and Related Topics for General Gaussian Processes","author":"Adler R. 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