{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T11:04:34Z","timestamp":1776769474159,"version":"3.51.2"},"reference-count":12,"publisher":"American Mathematical Society (AMS)","issue":"218","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    In a number of problems of mathematical physics and other fields stochastic differential equations are used to model certain phenomena. Often the solution of those problems can be obtained as a functional of the solution of some specific stochastic differential equation. Then we may use the idea of weak approximation to carry out numerical simulation. We analyze some complexity issues for a class of linear stochastic differential equations (Langevin type), which can be given by\n                    <disp-formula content-type=\"math\/mathml\">\n                      \\[\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"d upper X Subscript t Baseline equals minus alpha upper X Subscript t Baseline d t plus beta left-parenthesis t right-parenthesis d upper W Subscript t Baseline comma upper X 0 colon equals 0 comma\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>d<\/mml:mi>\n                            <mml:msub>\n                              <mml:mi>X<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>t<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mi>\n                              \u03b1\n                              \n                            <\/mml:mi>\n                            <mml:msub>\n                              <mml:mi>X<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>t<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mi>d<\/mml:mi>\n                            <mml:mi>t<\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mi>\n                              \u03b2\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>t<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mi>d<\/mml:mi>\n                            <mml:msub>\n                              <mml:mi>W<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>t<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mspace width=\"1em\"\/>\n                            <mml:msub>\n                              <mml:mi>X<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>0<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo>:=<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">dX_{t}=-\\alpha X_{t}dt+\\beta (t)dW_{t}, \\quad X_{0}:= 0,<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                      \\]\n                    <\/disp-formula>\n                    where\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"alpha greater-than 0\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03b1\n                              \n                            <\/mml:mi>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\alpha &gt;0<\/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=\"beta colon left-bracket 0 comma upper T right-bracket right-arrow double-struck upper R\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03b2\n                              \n                            <\/mml:mi>\n                            <mml:mo>:<\/mml:mo>\n                            <mml:mo stretchy=\"false\">[<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>T<\/mml:mi>\n                            <mml:mo stretchy=\"false\">]<\/mml:mo>\n                            <mml:mo stretchy=\"false\">\n                              \u2192\n                              \n                            <\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"double-struck\">R<\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\beta : [0,T]\\to \\mathbb {R}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . It turns out that for a class of input data which are not more than Lipschitz continuous the explicit Euler scheme gives rise to an optimal (by order) numerical method. Then we study numerical phenomena which occur when switching from (real) Monte Carlo simulation to quasi\u2013Monte Carlo simulation, which is the case when we carry out the simulation on computers. It will easily be seen that completely uniformly distributed sequences yield good substitutes for random variates, while not all uniformly distributed (mod 1) sequences are suited. In fact we provide necessary conditions on a sequence in order to serve quasi\u2013Monte Carlo purposes. This condition is expressed in terms of the measure of well-distributions. Numerical examples complement the theoretical analysis.\n                  <\/p>","DOI":"10.1090\/s0025-5718-97-00820-x","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:44Z","timestamp":1027707284000},"page":"573-589","source":"Crossref","is-referenced-by-count":13,"title":["On quasi-Monte Carlo simulation of stochastic differential equations"],"prefix":"10.1090","volume":"66","author":[{"given":"Norbert","family":"Hofmann","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Peter","family":"Math\u00e9","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[1997]]},"reference":[{"key":"1","doi-asserted-by":"crossref","unstructured":"N. N. Chentsov. Pseudo\u2013random numbers for modeling Markov chains. Zh. Vychisl. Mat. i Mat. Fiz., 7:632 \u2013 643, 1967.","DOI":"10.1016\/0041-5553(67)90041-9"},{"key":"2","doi-asserted-by":"publisher","first-page":"28","DOI":"10.2307\/2003733","article-title":"Deterministic simulation of random processes","volume":"17","author":"Franklin, Joel N.","year":"1963","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"3","first-page":"103","article-title":"L\u00f6sung von Integralgleichungen mittels zahlentheoretischer Methoden. I","volume":"171","author":"Hlawka, Edmund","year":"1962","journal-title":"\\\"{O}sterreich. Akad. Wiss. Math.-Natur. Kl. S.-B. II","ISSN":"https:\/\/id.crossref.org\/issn\/0029-8816","issn-type":"print"},{"key":"4","unstructured":"N. Hofmann. Beitr\u00e4ge zur schwachen Approximation stochastischer Differentialgleichungen. Dissertation, HU Berlin, 1995."},{"key":"5","series-title":"Graduate Texts in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4684-0302-2","volume-title":"Brownian motion and stochastic calculus","volume":"113","author":"Karatzas, Ioannis","year":"1988","ISBN":"https:\/\/id.crossref.org\/isbn\/0387965351"},{"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"},{"key":"7","volume-title":"The art of computer programming. Vol. 2: Seminumerical algorithms","author":"Knuth, Donald E.","year":"1969"},{"key":"8","series-title":"Pure and Applied Mathematics","volume-title":"Uniform distribution of sequences","author":"Kuipers, L.","year":"1974"},{"key":"9","unstructured":"P. Math\u00e9. Approximation theory of Monte Carlo methods. Habilitation thesis, 1994."},{"key":"10","series-title":"Mathematics and its Applications","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-015-8455-5","volume-title":"Numerical integration of stochastic differential equations","volume":"313","author":"Milstein, G. N.","year":"1995","ISBN":"https:\/\/id.crossref.org\/isbn\/079233213X"},{"issue":"6","key":"11","doi-asserted-by":"publisher","first-page":"957","DOI":"10.1090\/S0002-9904-1978-14532-7","article-title":"Quasi-Monte Carlo methods and pseudo-random numbers","volume":"84","author":"Niederreiter, Harald","year":"1978","journal-title":"Bull. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9904","issn-type":"print"},{"key":"12","series-title":"CBMS-NSF Regional Conference Series in Applied Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611970081","volume-title":"Random number generation and quasi-Monte Carlo methods","volume":"63","author":"Niederreiter, Harald","year":"1992","ISBN":"https:\/\/id.crossref.org\/isbn\/0898712955"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/1997-66-218\/S0025-5718-97-00820-X\/S0025-5718-97-00820-X.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/1997-66-218\/S0025-5718-97-00820-X\/S0025-5718-97-00820-X.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T21:26:46Z","timestamp":1776720406000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/1997-66-218\/S0025-5718-97-00820-X\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1997]]},"references-count":12,"journal-issue":{"issue":"218","published-print":{"date-parts":[[1997,4]]}},"alternative-id":["S0025-5718-97-00820-X"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-97-00820-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":[[1997]]}}}