{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,5]],"date-time":"2026-03-05T20:50:27Z","timestamp":1772743827845,"version":"3.50.1"},"reference-count":24,"publisher":"Walter de Gruyter GmbH","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2018,12,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>We explore different methods of solving systems of stochastic differential equations by first implementing the Euler\u2013Maruyama and Milstein methods with a Monte Carlo simulation on a CPU. The performance of the methods is significantly improved through the recently developed antithetic multilevel Monte Carlo estimator, which yields a computation complexity of <jats:inline-formula id=\"j_mcma-2018-2025_ineq_9999_w2aab3b7b6b1b6b1aab1c16b1b1Aa\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mi>\ud835\udcaa<\/m:mi>\n                              <m:mo>\u2062<\/m:mo>\n                              <m:mrow>\n                                 <m:mo>(<\/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>)<\/m:mo>\n                              <\/m:mrow>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_mcma-2018-2025_eq_0177.png\"\/>\n                        <jats:tex-math>{\\mathcal{O}(\\epsilon^{-2})}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> root-mean-square error and does so without the approximation of L\u00e9vy areas. Further improvements in performance are gained by moving the algorithms to a GPU - first on a single device and then on a multi-GPU cluster. Our GPU implementation of the antithetic multilevel Monte Carlo displays a major speedup in computation when compared with many commonly used approaches in the literature. While our work is focused on the simulation of the stochastic volatility and interest rate model, it is easily extendable to other stochastic systems, and it is of particular interest to those with non-diagonal, non-commutative noise.<\/jats:p>","DOI":"10.1515\/mcma-2018-2025","type":"journal-article","created":{"date-parts":[[2018,10,30]],"date-time":"2018-10-30T09:27:16Z","timestamp":1540891636000},"page":"309-321","source":"Crossref","is-referenced-by-count":4,"title":["On the implementation of multilevel Monte Carlo simulation of the stochastic volatility and interest rate model using multi-GPU clusters"],"prefix":"10.1515","volume":"24","author":[{"given":"Harold A.","family":"Lay","sequence":"first","affiliation":[{"name":"Thompson Machinery Commerce Corporation , 1245 Bridgestone Blvd. , LaVergne , TN 37086 , USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zane","family":"Colgin","sequence":"additional","affiliation":[{"name":"Applied Physics Laboratory , The Johns Hopkins University , Laurel , MD 20723 , USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Viktor","family":"Reshniak","sequence":"additional","affiliation":[{"name":"Oak Ridge National Laboratory , Computer Science and Mathematics Division , Oak Ridge , TN 37831 , USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Abdul Q.\u2009M.","family":"Khaliq","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences and Center for Computational Science , Middle Tennessee State University , 1301 East Main Street , Murfreesboro , TN 37130 , USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2018,10,30]]},"reference":[{"key":"2023040101333956178_j_mcma-2018-2025_ref_001_w2aab3b7b6b1b6b1ab1b5b1Aa","doi-asserted-by":"crossref","unstructured":"D. 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Cameron,\nThe maximum rate of convergence of discrete approximations for stochastic differential equations,\nStochastic Differential Systems (Vilnius 1978),\nLecture Notes in Control and Inform. Sci. 25,\nSpringer, Berlin (1980), 162\u2013171.","DOI":"10.1007\/BFb0004007"},{"key":"2023040101333956178_j_mcma-2018-2025_ref_007_w2aab3b7b6b1b6b1ab1b5b7Aa","doi-asserted-by":"crossref","unstructured":"A. M.  Dimits, B. I.  Cohen, R. E.  Caflisch, M. S.  Rosin and L. F.  Ricketson,\nHigher-order time integration of Coulomb collisions in a plasma using Langevin equations,\nJ. Comput. Phys. 242 (2013), 561\u2013580.\n10.1016\/j.jcp.2013.01.038","DOI":"10.1016\/j.jcp.2013.01.038"},{"key":"2023040101333956178_j_mcma-2018-2025_ref_008_w2aab3b7b6b1b6b1ab1b5b8Aa","doi-asserted-by":"crossref","unstructured":"J. G.  Gaines and T. J.  Lyons,\nRandom generation of stochastic area integrals,\nSIAM J. Appl. 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