{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T22:47:31Z","timestamp":1776725251373,"version":"3.51.2"},"reference-count":10,"publisher":"American Mathematical Society (AMS)","issue":"234","license":[{"start":{"date-parts":[[2001,3,2]],"date-time":"2001-03-02T00:00:00Z","timestamp":983491200000},"content-version":"am","delay-in-days":365,"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 complexity of approximating stochastic integrals with error\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"epsilon\">\n                        <mml:semantics>\n                          <mml:mi>\n                            \u03b5\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\varepsilon<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    for various classes of functions. For Ito integration, we show that the complexity is of order\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"epsilon Superscript negative 1\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>\n                              \u03b5\n                              \n                            <\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\n                                \u2212\n                                \n                              <\/mml:mo>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:mrow>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">\\varepsilon ^{-1}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , even for classes of very smooth functions. The lower bound is obtained by showing that Ito integration is not easier than Lebesgue integration in the average case setting with the Wiener measure. The upper bound is obtained by the Milstein algorithm, which is almost optimal in the considered classes of functions. The Milstein algorithm uses the values of the Brownian motion and the integrand. It is bilinear in these values and is very easy to implement. For Stratonovich integration, we show that the complexity depends on the smoothness of the integrand and may be much smaller than the complexity of Ito integration.\n                  <\/p>","DOI":"10.1090\/s0025-5718-00-01214-x","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:17:46Z","timestamp":1027707466000},"page":"685-698","source":"Crossref","is-referenced-by-count":15,"title":["On the complexity of stochastic integration"],"prefix":"10.1090","volume":"70","author":[{"given":"G.","family":"Wasilkowski","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"H.","family":"Wo\u017aniakowski","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2000,3,2]]},"reference":[{"key":"1","series-title":"Probability and its Applications","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-4480-6","volume-title":"Introduction to stochastic integration","author":"Chung, K. L.","year":"1990","ISBN":"https:\/\/id.crossref.org\/isbn\/0817633863","edition":"2"},{"key":"2","unstructured":"O. Faure, Simulation du mouvement brownien et des diffusions, These ENPC, Paris, 1992."},{"key":"3","unstructured":"N. Hofmann, T. M\u00fcller-Gronbach, and K. Ritter, Optimal approximation of stochastic differential equations by adaptive step-size control, Math. Comp., Posted on May 20, 1999, PII S 0025-5718(99)01177-1 (to appear in print)."},{"key":"4","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":"4","key":"5","doi-asserted-by":"publisher","first-page":"641","DOI":"10.1216\/RMJ-1986-16-4-641","article-title":"Approximation of linear operators on a Wiener space","volume":"16","author":"Lee, D.","year":"1986","journal-title":"Rocky Mountain J. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0035-7596","issn-type":"print"},{"key":"6","series-title":"Universitext","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-02847-6","volume-title":"Stochastic differential equations","author":"\u00d8ksendal, Bernt","year":"1992","ISBN":"https:\/\/id.crossref.org\/isbn\/3540533354","edition":"3"},{"key":"7","series-title":"Computer Science and Scientific Computing","isbn-type":"print","volume-title":"Information-based complexity","author":"Traub, J. F.","year":"1988","ISBN":"https:\/\/id.crossref.org\/isbn\/0126975450"},{"issue":"3","key":"8","doi-asserted-by":"publisher","first-page":"204","DOI":"10.1016\/0885-064X(86)90002-6","article-title":"Information of varying cardinality","volume":"2","author":"Wasilkowski, G. W.","year":"1986","journal-title":"J. Complexity","ISSN":"https:\/\/id.crossref.org\/issn\/0885-064X","issn-type":"print"},{"issue":"4","key":"9","doi-asserted-by":"publisher","first-page":"457","DOI":"10.1016\/0885-064X(89)90020-4","article-title":"Mixed settings for linear problems","volume":"5","author":"Wasilkowski, G. W.","year":"1989","journal-title":"J. Complexity","ISSN":"https:\/\/id.crossref.org\/issn\/0885-064X","issn-type":"print"},{"key":"10","unstructured":"G. W. Wasilkowski and H. Wo\u017aniakowski, Weighted approximation over \u211d, to appear in J. Approx. Theory."}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2001-70-234\/S0025-5718-00-01214-X\/S0025-5718-00-01214-X.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2001-70-234\/S0025-5718-00-01214-X\/S0025-5718-00-01214-X.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T22:35:15Z","timestamp":1776724515000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2001-70-234\/S0025-5718-00-01214-X\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2000,3,2]]},"references-count":10,"journal-issue":{"issue":"234","published-print":{"date-parts":[[2001,4]]}},"alternative-id":["S0025-5718-00-01214-X"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-00-01214-x","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2000,3,2]]}}}