{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T22:44:56Z","timestamp":1776725096448,"version":"3.51.2"},"reference-count":20,"publisher":"American Mathematical Society (AMS)","issue":"231","license":[{"start":{"date-parts":[[2000,5,20]],"date-time":"2000-05-20T00:00:00Z","timestamp":958780800000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    We study the pathwise (strong) approximation of scalar stochastic differential equations with respect to the global error in the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L 2\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>L<\/mml:mi>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">L_2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -norm. For equations with additive noise we establish a sharp lower error bound in the class of arbitrary methods that use a fixed number of observations of the driving Brownian motion. As a consequence, higher order methods do not exist if the global error is analyzed. We introduce an adaptive step-size control for the Euler scheme which performs asymptotically optimally. In particular, the new method is more efficient than an equidistant discretization. This superiority is confirmed in simulation experiments for equations with additive noise, as well as for general scalar equations.\n                  <\/p>","DOI":"10.1090\/s0025-5718-99-01177-1","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:13:45Z","timestamp":1027707225000},"page":"1017-1034","source":"Crossref","is-referenced-by-count":39,"title":["Optimal approximation of stochastic differential equations by adaptive step-size control"],"prefix":"10.1090","volume":"69","author":[{"given":"Norbert","family":"Hofmann","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Thomas","family":"M\u00fcller-Gronbach","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Klaus","family":"Ritter","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[1999,5,20]]},"reference":[{"key":"1","series-title":"Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics","isbn-type":"print","volume-title":"Numerical methods for stochastic processes","author":"Bouleau, Nicolas","year":"1994","ISBN":"https:\/\/id.crossref.org\/isbn\/0471546410"},{"issue":"3-4","key":"2","doi-asserted-by":"publisher","first-page":"211","DOI":"10.1080\/17442509608834090","article-title":"Exact convergence rate of the Euler-Maruyama scheme, with application to sampling design","volume":"59","author":"Cambanis, Stamatis","year":"1996","journal-title":"Stochastics Stochastics Rep.","ISSN":"https:\/\/id.crossref.org\/issn\/1045-1129","issn-type":"print"},{"key":"3","isbn-type":"print","first-page":"162","article-title":"The maximum rate of convergence of discrete approximations for stochastic differential equations","author":"Clark, J. 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Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1399","issn-type":"print"},{"key":"6","series-title":"Applications of Mathematics (New York)","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-12616-5","volume-title":"Numerical solution of stochastic differential equations","volume":"23","author":"Kloeden, Peter E.","year":"1992","ISBN":"https:\/\/id.crossref.org\/isbn\/3540540628"},{"issue":"1","key":"7","doi-asserted-by":"publisher","first-page":"47","DOI":"10.1006\/jcom.1996.0006","article-title":"About widths of Wiener space in the \ud835\udc3f_{\ud835\udc5e}-norm","volume":"12","author":"Maiorov, V. E.","year":"1996","journal-title":"J. Complexity","ISSN":"https:\/\/id.crossref.org\/issn\/0885-064X","issn-type":"print"},{"key":"#cr-split#-8.1","doi-asserted-by":"crossref","unstructured":"Mauthner, S. (1998). Step size control in the numerical solution of stochastic differential equations. 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Inference","ISSN":"https:\/\/id.crossref.org\/issn\/0378-3758","issn-type":"print"},{"issue":"2","key":"11","doi-asserted-by":"publisher","first-page":"227","DOI":"10.1080\/17442509008833616","article-title":"An efficient approximation for stochastic differential equations on the partition of symmetrical first passage times","volume":"29","author":"Newton, Nigel J.","year":"1990","journal-title":"Stochastics Stochastics Rep.","ISSN":"https:\/\/id.crossref.org\/issn\/1045-1129","issn-type":"print"},{"issue":"1","key":"12","doi-asserted-by":"publisher","first-page":"15","DOI":"10.1006\/jcom.1993.1003","article-title":"Some complexity results for zero finding for univariate functions","volume":"9","author":"Novak, Erich","year":"1993","journal-title":"J. Complexity","ISSN":"https:\/\/id.crossref.org\/issn\/0885-064X","issn-type":"print"},{"key":"13","doi-asserted-by":"crossref","unstructured":"Ritter, K. (1999). Average Case Analysis of Numerical Problems. Lect. 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