{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T22:37:35Z","timestamp":1776724655448,"version":"3.51.2"},"reference-count":11,"publisher":"American Mathematical Society (AMS)","issue":"221","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    This paper investigates the rate of convergence of an alternative approximation method for stochastic differential equations. The rates of convergence of the one-step and multi-step approximation errors are proved to be\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper O left-parenthesis left-parenthesis normal upper Delta t right-parenthesis squared right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>O<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi mathvariant=\"normal\">\n                              \u0394\n                              \n                            <\/mml:mi>\n                            <mml:mi>t<\/mml:mi>\n                            <mml:msup>\n                              <mml:mo stretchy=\"false\">)<\/mml:mo>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">O((\\Delta t)^2)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper O left-parenthesis normal upper Delta t right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>O<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi mathvariant=\"normal\">\n                              \u0394\n                              \n                            <\/mml:mi>\n                            <mml:mi>t<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">O(\\Delta t)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    in the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L Subscript p\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>L<\/mml:mi>\n                            <mml:mi>p<\/mml:mi>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">L_p<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    sense respectively, where\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"normal upper Delta t\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"normal\">\n                              \u0394\n                              \n                            <\/mml:mi>\n                            <mml:mi>t<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\Delta t<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is discrete time interval. The rate of convergence of the one-step approximation error is improved as compared with methods assuming the value of Brownian motion to be known only at discrete time. Through numerical experiments, the rate of convergence of the multi-step approximation error is seen to be much faster than in the conventional method.\n                  <\/p>","DOI":"10.1090\/s0025-5718-98-00888-6","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:28Z","timestamp":1027707268000},"page":"287-298","source":"Crossref","is-referenced-by-count":21,"title":["Approximation of continuous time stochastic processes by a local linearization method"],"prefix":"10.1090","volume":"67","author":[{"given":"Isao","family":"Shoji","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[1998]]},"reference":[{"issue":"180","key":"1","doi-asserted-by":"publisher","first-page":"523","DOI":"10.2307\/2008326","article-title":"Numerical solution of stochastic differential equations with constant diffusion coefficients","volume":"49","author":"Chang, Chien Cheng","year":"1987","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"2","isbn-type":"print","first-page":"162","article-title":"The maximum rate of convergence of discrete approximations for stochastic differential equations","author":"Clark, J. M. C.","year":"1980","ISBN":"https:\/\/id.crossref.org\/isbn\/3540104984"},{"key":"3","series-title":"Graduate Texts in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-0949-2","volume-title":"Brownian motion and stochastic calculus","volume":"113","author":"Karatzas, Ioannis","year":"1991","ISBN":"https:\/\/id.crossref.org\/isbn\/0387976558","edition":"2"},{"key":"4","series-title":"Research and Exposition in Mathematics","isbn-type":"print","volume-title":"Parameter estimation for stochastic processes","volume":"6","author":"Kutoyants, Yu. 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Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1399","issn-type":"print"},{"key":"8","doi-asserted-by":"crossref","unstructured":"Ozaki, T., Statistical identification of storage models with application to stochastic hydrology, Water Resources Bulletin 21 (1985), 663\u2013675.","DOI":"10.1111\/j.1752-1688.1985.tb05381.x"},{"issue":"3","key":"9","doi-asserted-by":"publisher","first-page":"604","DOI":"10.1137\/0719041","article-title":"Numerical treatment of stochastic differential equations","volume":"19","author":"R\u00fcmelin, W.","year":"1982","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"10","unstructured":"Shoji, I. and Ozaki, T., Estimation for nonlinear stochastic differential equations by a local linearization method, forthcoming in Stochastic Analysis and Applications."},{"issue":"2","key":"11","doi-asserted-by":"publisher","first-page":"220","DOI":"10.1016\/0047-259X(92)90068-Q","article-title":"Estimation for diffusion processes from discrete observation","volume":"41","author":"Yoshida, Nakahiro","year":"1992","journal-title":"J. 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