{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,18]],"date-time":"2026-04-18T04:31:36Z","timestamp":1776486696847,"version":"3.51.2"},"reference-count":12,"publisher":"Walter de Gruyter GmbH","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2016,9,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>In this paper, we are interested in the strong convergence properties of the Ninomiya\u2013Victoir scheme which is known to exhibit weak convergence with order 2. We prove strong convergence with order <jats:inline-formula id=\"j_mcma-2016-0109_ineq_9999_w2aab2b8d620b1b7b1aab1c13b1b1Aa\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mn>1<\/m:mn>\n                              <m:mo>\/<\/m:mo>\n                              <m:mn>2<\/m:mn>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" content-type=\"j_mcma-2016-0109_ineq_9999\" xlink:href=\"graphic\/j_mcma-2016-0109_eq_mi306.png\"\/>\n                        <jats:tex-math>${1\/2}$<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>. This study is aimed at analysing the use of this scheme either at each level or only at the finest level of a multilevel Monte Carlo estimator: indeed, the variance of a multilevel Monte Carlo estimator is related to the strong error between the two schemes used on the coarse and fine grids at each level.\nRecently, Giles and Szpruch proposed a scheme permitting to construct a multilevel Monte Carlo estimator achieving the optimal complexity <jats:inline-formula id=\"j_mcma-2016-0109_ineq_9998_w2aab2b8d620b1b7b1aab1c13b1b3Aa\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mi>O<\/m:mi>\n                              <m:mo>\u2062<\/m:mo>\n                              <m:mrow>\n                                 <m:mo stretchy=\"false\">(<\/m:mo>\n                                 <m:msup>\n                                    <m:mi>\u03f5<\/m:mi>\n                                    <m:mrow>\n                                       <m:mo>-<\/m:mo>\n                                       <m:mn>2<\/m:mn>\n                                    <\/m:mrow>\n                                 <\/m:msup>\n                                 <m:mo stretchy=\"false\">)<\/m:mo>\n                              <\/m:mrow>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" content-type=\"j_mcma-2016-0109_ineq_9998\" xlink:href=\"graphic\/j_mcma-2016-0109_eq_mi384.png\"\/>\n                        <jats:tex-math>${O(\\epsilon^{-2})}$<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> for the precision \u03f5.\nIn the same spirit, we propose a modified Ninomiya\u2013Victoir scheme, which may be strongly coupled with order 1 to the Giles\u2013Szpruch scheme at the finest level of a multilevel Monte Carlo estimator. Numerical experiments show that this choice improves the efficiency, since the order 2 of weak convergence of the Ninomiya\u2013Victoir scheme permits to reduce the number of discretisation levels.<\/jats:p>","DOI":"10.1515\/mcma-2016-0109","type":"journal-article","created":{"date-parts":[[2016,6,30]],"date-time":"2016-06-30T10:02:40Z","timestamp":1467280960000},"page":"197-228","source":"Crossref","is-referenced-by-count":5,"title":["Ninomiya\u2013Victoir scheme: Strong convergence, antithetic version and application to multilevel estimators"],"prefix":"10.1515","volume":"22","author":[{"given":"Anis","family":"Al Gerbi","sequence":"first","affiliation":[{"name":"Universit\u00e9 Paris-Est, Cermics (ENPC), INRIA, F-77455, Marne-la-Vall\u00e9e, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Benjamin","family":"Jourdain","sequence":"additional","affiliation":[{"name":"Universit\u00e9 Paris-Est, Cermics (ENPC), INRIA, F-77455, Marne-la-Vall\u00e9e, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Emmanuelle","family":"Cl\u00e9ment","sequence":"additional","affiliation":[{"name":"Universit\u00e9 Paris-Est, LAMA (UMR 8050), UPEMLV, UPEC, CNRS, F-77454, Marne-la-Vall\u00e9e, France"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2016,6,30]]},"reference":[{"key":"2023040101594999053_j_mcma-2016-0109_ref_001_w2aab2b8d620b1b7b1ab2ab1Aa","doi-asserted-by":"crossref","unstructured":"Alfonsi A.,\nAffine Diffusions and Related Processes: Simulation, Theory and Applications,\nBocconi Springer Ser. 6,\nSpringer, Cham, 2015.","DOI":"10.1007\/978-3-319-05221-2"},{"key":"2023040101594999053_j_mcma-2016-0109_ref_002_w2aab2b8d620b1b7b1ab2ab2Aa","doi-asserted-by":"crossref","unstructured":"Al Gerbi A., Jourdain B. and Cl\u00e9ment E.,\nNinomiya\u2013Victoir scheme: Strong convergence, antithetic version and application to multilevel estimators,\npreprint 2015, http:\/\/arxiv.org\/abs\/1508.06492.","DOI":"10.1515\/mcma-2016-0109"},{"key":"2023040101594999053_j_mcma-2016-0109_ref_003_w2aab2b8d620b1b7b1ab2ab3Aa","doi-asserted-by":"crossref","unstructured":"Debrabant K. and R\u00f6\u00dfler A.,\nOn the acceleration of the multi-level Monte Carlo method,\nJ. 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