{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T20:43:55Z","timestamp":1776804235032,"version":"3.51.2"},"reference-count":16,"publisher":"American Mathematical Society (AMS)","issue":"314","license":[{"start":{"date-parts":[[2019,2,19]],"date-time":"2019-02-19T00:00:00Z","timestamp":1550534400000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["71601189"],"award-info":[{"award-number":["71601189"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    This paper studies randomized quasi-Monte Carlo (QMC) sampling for discontinuous integrands having singularities along the boundary of the unit cube\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-bracket 0 comma 1 right-bracket Superscript d\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo stretchy=\"false\">[<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:msup>\n                              <mml:mo stretchy=\"false\">]<\/mml:mo>\n                              <mml:mi>d<\/mml:mi>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">[0,1]^d<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . Both discontinuities and singularities are extremely common in the pricing and hedging of financial derivatives and have a tremendous impact on the accuracy of QMC. It was previously known that the root mean square error of randomized QMC is only\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"o left-parenthesis n Superscript 1 slash 2 Baseline right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>o<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>n<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>1<\/mml:mn>\n                                <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                  <mml:mo>\/<\/mml:mo>\n                                <\/mml:mrow>\n                                <mml:mn>2<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">o(n^{1\/2})<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    for discontinuous functions with singularities. We find that under some mild conditions, randomized QMC yields an expected error of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper O left-parenthesis n Superscript negative 1 slash 2 minus 1 slash left-parenthesis 4 d minus 2 right-parenthesis plus epsilon Baseline right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>O<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>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 class=\"MJX-TeXAtom-ORD\">\n                                  <mml:mo>\/<\/mml:mo>\n                                <\/mml:mrow>\n                                <mml:mn>2<\/mml:mn>\n                                <mml:mo>\n                                  \u2212\n                                  \n                                <\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                                <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                  <mml:mo>\/<\/mml:mo>\n                                <\/mml:mrow>\n                                <mml:mo stretchy=\"false\">(<\/mml:mo>\n                                <mml:mn>4<\/mml:mn>\n                                <mml:mi>d<\/mml:mi>\n                                <mml:mo>\n                                  \u2212\n                                  \n                                <\/mml:mo>\n                                <mml:mn>2<\/mml:mn>\n                                <mml:mo stretchy=\"false\">)<\/mml:mo>\n                                <mml:mo>+<\/mml:mo>\n                                <mml:mi>\n                                  \u03f5\n                                  \n                                <\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">O(n^{-1\/2-1\/(4d-2)+\\epsilon })<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    for arbitrarily small\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"epsilon greater-than 0\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03f5\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\">\\epsilon &gt;0<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . Moreover, one can get a better rate if the boundary of discontinuities is parallel to some coordinate axes. As a by-product, we find that the expected error rate attains\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper O left-parenthesis n Superscript negative 1 plus epsilon Baseline right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>O<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>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:mo>+<\/mml:mo>\n                                <mml:mi>\n                                  \u03f5\n                                  \n                                <\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">O(n^{-1+\\epsilon })<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    if the discontinuities are QMC-friendly, in the sense that all the discontinuity boundaries are parallel to coordinate axes. The results can be used to assess the QMC accuracy for some typical problems from financial engineering.\n                  <\/p>","DOI":"10.1090\/mcom\/3324","type":"journal-article","created":{"date-parts":[[2017,9,13]],"date-time":"2017-09-13T10:40:34Z","timestamp":1505299234000},"page":"2857-2870","source":"Crossref","is-referenced-by-count":7,"title":["Quasi-Monte Carlo for discontinuous integrands with singularities along the boundary of the unit cube"],"prefix":"10.1090","volume":"87","author":[{"given":"Zhijian","family":"He","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2018,2,19]]},"reference":[{"issue":"4","key":"1","doi-asserted-by":"publisher","first-page":"727","DOI":"10.1007\/s00208-008-0254-z","article-title":"Outer Minkowski content for some classes of closed sets","volume":"342","author":"Ambrosio, Luigi","year":"2008","journal-title":"Math. 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