{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,3]],"date-time":"2026-06-03T06:42:09Z","timestamp":1780468929326,"version":"3.54.1"},"reference-count":42,"publisher":"American Mathematical Society (AMS)","issue":"346","license":[{"start":{"date-parts":[[2024,11,2]],"date-time":"2024-11-02T00:00:00Z","timestamp":1730505600000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100008419","name":"Institut Louis Bachelier","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100008419","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n In this work, we develop a multifactor approximation for\n \n \n \n d<\/mml:mi>\n d<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -dimensional Stochastic Volterra Equations (SVE) with Lipschitz coefficients and kernels of completely monotone type that may be singular. First, we prove an\n \n \n \n \n L<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n L^2<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -estimation between two SVEs with different kernels, which provides a quantification of the error between the SVE and any multifactor Stochastic Differential Equation (SDE) approximation. For the particular rough kernel case with Hurst parameter lying in\n \n \n \n \n (<\/mml:mo>\n 0<\/mml:mn>\n ,<\/mml:mo>\n 1<\/mml:mn>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n 2<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n (0,1\/2)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , we propose various approximating multifactor kernels, state their rates of convergence and illustrate their efficiency for the rough Bergomi model. Second, we study a Euler discretization of the multifactor SDE and establish a convergence result towards the SVE that is uniform with respect to the approximating multifactor kernels. These obtained results lead us to build a new multifactor Euler scheme that reduces significantly the computational cost in an asymptotic way compared to the Euler scheme for SVEs. Finally, we show that our multifactor Euler scheme outperforms the Euler scheme for SVEs for option pricing in the rough Heston model.\n <\/p>","DOI":"10.1090\/mcom\/3911","type":"journal-article","created":{"date-parts":[[2023,9,27]],"date-time":"2023-09-27T10:34:30Z","timestamp":1695810870000},"page":"643-677","source":"Crossref","is-referenced-by-count":16,"title":["Approximation of Stochastic Volterra Equations with kernels of completely monotone type"],"prefix":"10.1090","volume":"93","author":[{"given":"Aur\u00e9lien","family":"Alfonsi","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ahmed","family":"Kebaier","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2023,11,2]]},"reference":[{"issue":"3","key":"1","doi-asserted-by":"publisher","first-page":"1583","DOI":"10.3150\/20-bej1284","article-title":"Weak existence and uniqueness for affine stochastic Volterra equations with \ud835\udc3f\u00b9-kernels","volume":"27","author":"Abi Jaber, Eduardo","year":"2021","journal-title":"Bernoulli","ISSN":"https:\/\/id.crossref.org\/issn\/1350-7265","issn-type":"print"},{"issue":"2","key":"2","doi-asserted-by":"publisher","first-page":"309","DOI":"10.1137\/18M1170236","article-title":"Multifactor approximation of rough volatility models","volume":"10","author":"Abi Jaber, Eduardo","year":"2019","journal-title":"SIAM J. Financial Math."},{"issue":"269","key":"3","doi-asserted-by":"publisher","first-page":"209","DOI":"10.1090\/S0025-5718-09-02252-2","article-title":"High order discretization schemes for the CIR process: application to affine term structure and Heston models","volume":"79","author":"Alfonsi, Aur\u00e9lien","year":"2010","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"6","key":"4","doi-asserted-by":"publisher","first-page":"887","DOI":"10.1080\/14697688.2015.1099717","article-title":"Pricing under rough volatility","volume":"16","author":"Bayer, Christian","year":"2016","journal-title":"Quant. Finance","ISSN":"https:\/\/id.crossref.org\/issn\/1469-7688","issn-type":"print"},{"issue":"5","key":"5","doi-asserted-by":"publisher","first-page":"779","DOI":"10.1080\/14697688.2018.1529420","article-title":"Short-time near-the-money skew in rough fractional volatility models","volume":"19","author":"Bayer, C.","year":"2019","journal-title":"Quant. Finance","ISSN":"https:\/\/id.crossref.org\/issn\/1469-7688","issn-type":"print"},{"issue":"1","key":"6","doi-asserted-by":"publisher","first-page":"393","DOI":"10.1016\/j.jmaa.2017.09.002","article-title":"Variance reduction for discretised diffusions via regression","volume":"458","author":"Belomestny, Denis","year":"2018","journal-title":"J. Math. Anal. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0022-247X","issn-type":"print"},{"issue":"1","key":"7","doi-asserted-by":"publisher","first-page":"211","DOI":"10.1214\/13-AAP993","article-title":"Central limit theorem for the multilevel Monte Carlo Euler method","volume":"25","author":"Ben Alaya, Mohamed","year":"2015","journal-title":"Ann. Appl. Probab.","ISSN":"https:\/\/id.crossref.org\/issn\/1050-5164","issn-type":"print"},{"issue":"4","key":"8","doi-asserted-by":"publisher","first-page":"931","DOI":"10.1007\/s00780-017-0335-5","article-title":"Hybrid scheme for Brownian semistationary processes","volume":"21","author":"Bennedsen, Mikkel","year":"2017","journal-title":"Finance Stoch.","ISSN":"https:\/\/id.crossref.org\/issn\/0949-2984","issn-type":"print"},{"issue":"3","key":"9","first-page":"187","article-title":"Volterra equations with It\u00f4 integrals. I","volume":"2","author":"Berger, Marc A.","year":"1980","journal-title":"J. Integral Equations","ISSN":"https:\/\/id.crossref.org\/issn\/0163-5549","issn-type":"print"},{"issue":"4","key":"10","first-page":"319","article-title":"Volterra equations with It\u00f4 integrals. II","volume":"2","author":"Berger, Marc A.","year":"1980","journal-title":"J. Integral Equations","ISSN":"https:\/\/id.crossref.org\/issn\/0163-5549","issn-type":"print"},{"key":"11","doi-asserted-by":"publisher","first-page":"95","DOI":"10.1214\/ECP.v3-998","article-title":"Fractional Brownian motion and the Markov property","volume":"3","author":"Carmona, Philippe","year":"1998","journal-title":"Electron. Comm. Probab.","ISSN":"https:\/\/id.crossref.org\/issn\/1083-589X","issn-type":"print"},{"issue":"1-2","key":"12","doi-asserted-by":"publisher","first-page":"161","DOI":"10.1023\/A:1009999518898","article-title":"Approximation of some Gaussian processes","volume":"3","author":"Carmona, Philippe","year":"2000","journal-title":"Stat. Inference Stoch. Process.","ISSN":"https:\/\/id.crossref.org\/issn\/1387-0874","issn-type":"print"},{"issue":"2","key":"13","doi-asserted-by":"publisher","first-page":"337","DOI":"10.1016\/0304-4149(94)00072-2","article-title":"Stochastic Volterra equations with singular kernels","volume":"56","author":"Cochran, W. George","year":"1995","journal-title":"Stochastic Process. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0304-4149","issn-type":"print"},{"key":"14","isbn-type":"print","first-page":"39","article-title":"Stochastic Volterra equations with singular kernels","author":"Coutin, L.","year":"2001","ISBN":"https:\/\/id.crossref.org\/isbn\/0817642463"},{"issue":"2","key":"15","doi-asserted-by":"publisher","first-page":"139","DOI":"10.1023\/A:1012628013041","article-title":"Regularity properties of some stochastic Volterra integrals with singular kernel","volume":"16","author":"Decreusefond, L.","year":"2002","journal-title":"Potential Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0926-2601","issn-type":"print"},{"issue":"1","key":"16","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1111\/mafi.12173","article-title":"The characteristic function of rough Heston models","volume":"29","author":"El Euch, Omar","year":"2019","journal-title":"Math. Finance","ISSN":"https:\/\/id.crossref.org\/issn\/0960-1627","issn-type":"print"},{"issue":"3","key":"17","doi-asserted-by":"publisher","first-page":"463","DOI":"10.1080\/14697688.2021.1999486","article-title":"Short-dated smile under rough volatility: asymptotics and numerics","volume":"22","author":"Friz, Peter K.","year":"2022","journal-title":"Quant. Finance","ISSN":"https:\/\/id.crossref.org\/issn\/1469-7688","issn-type":"print"},{"issue":"2","key":"18","doi-asserted-by":"publisher","first-page":"189","DOI":"10.1080\/14697688.2016.1197410","article-title":"Short-time at-the-money skew and rough fractional volatility","volume":"17","author":"Fukasawa, Masaaki","year":"2017","journal-title":"Quant. Finance","ISSN":"https:\/\/id.crossref.org\/issn\/1469-7688","issn-type":"print"},{"issue":"1","key":"19","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1080\/14697688.2020.1825781","article-title":"Volatility has to be rough","volume":"21","author":"Fukasawa, Masaaki","year":"2021","journal-title":"Quant. Finance","ISSN":"https:\/\/id.crossref.org\/issn\/1469-7688","issn-type":"print"},{"issue":"6","key":"20","doi-asserted-by":"publisher","first-page":"933","DOI":"10.1080\/14697688.2017.1393551","article-title":"Volatility is rough","volume":"18","author":"Gatheral, Jim","year":"2018","journal-title":"Quant. Finance","ISSN":"https:\/\/id.crossref.org\/issn\/1469-7688","issn-type":"print"},{"issue":"3","key":"21","doi-asserted-by":"publisher","first-page":"607","DOI":"10.1287\/opre.1070.0496","article-title":"Multilevel Monte Carlo path simulation","volume":"56","author":"Giles, Michael B.","year":"2008","journal-title":"Oper. Res.","ISSN":"https:\/\/id.crossref.org\/issn\/0030-364X","issn-type":"print"},{"key":"22","series-title":"Encyclopedia of Mathematics and its Applications","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511662805","volume-title":"Volterra integral and functional equations","volume":"34","author":"Gripenberg, G.","year":"1990","ISBN":"https:\/\/id.crossref.org\/isbn\/0521372895"},{"issue":"10","key":"23","doi-asserted-by":"publisher","first-page":"5567","DOI":"10.3934\/dcdsb.2020367","article-title":"Strong convergence rates for Markovian representations of fractional processes","volume":"26","author":"Harms, Philipp","year":"2021","journal-title":"Discrete Contin. Dyn. Syst. Ser. B","ISSN":"https:\/\/id.crossref.org\/issn\/1531-3492","issn-type":"print"},{"issue":"4","key":"24","doi-asserted-by":"publisher","first-page":"1185","DOI":"10.1016\/j.spa.2018.04.010","article-title":"Affine representations of fractional processes with applications in mathematical finance","volume":"129","author":"Harms, Philipp","year":"2019","journal-title":"Stochastic Process. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0304-4149","issn-type":"print"},{"key":"25","isbn-type":"print","volume-title":"Analysis of numerical methods","author":"Isaacson, Eugene","year":"1994","ISBN":"https:\/\/id.crossref.org\/isbn\/0486680290"},{"issue":"5","key":"26","doi-asserted-by":"publisher","first-page":"1687","DOI":"10.1214\/09-AAP595","article-title":"Robust adaptive importance sampling for normal random vectors","volume":"19","author":"Jourdain, Benjamin","year":"2009","journal-title":"Ann. Appl. Probab.","ISSN":"https:\/\/id.crossref.org\/issn\/1050-5164","issn-type":"print"},{"key":"27","isbn-type":"print","doi-asserted-by":"publisher","first-page":"69","DOI":"10.1007\/978-4-431-68450-3_4","article-title":"Approximation of expectation of diffusion processes based on Lie algebra and Malliavin calculus","author":"Kusuoka, Shigeo","year":"2004","ISBN":"https:\/\/id.crossref.org\/isbn\/443120315X"},{"issue":"3","key":"28","doi-asserted-by":"publisher","first-page":"1029","DOI":"10.1214\/09-AAP650","article-title":"Unconstrained recursive importance sampling","volume":"20","author":"Lemaire, Vincent","year":"2010","journal-title":"Ann. Appl. Probab.","ISSN":"https:\/\/id.crossref.org\/issn\/1050-5164","issn-type":"print"},{"issue":"4","key":"29","doi-asserted-by":"publisher","first-page":"2643","DOI":"10.3150\/16-BEJ822","article-title":"Multilevel Richardson-Romberg extrapolation","volume":"23","author":"Lemaire, Vincent","year":"2017","journal-title":"Bernoulli","ISSN":"https:\/\/id.crossref.org\/issn\/1350-7265","issn-type":"print"},{"issue":"6","key":"30","doi-asserted-by":"publisher","first-page":"1780","DOI":"10.1137\/S0036139992236220","article-title":"Variance reduction for simulated diffusions","volume":"54","author":"Newton, Nigel J.","year":"1994","journal-title":"SIAM J. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1399","issn-type":"print"},{"issue":"1-2","key":"31","doi-asserted-by":"publisher","first-page":"107","DOI":"10.1080\/13504860701413958","article-title":"Weak approximation of stochastic differential equations and application to derivative pricing","volume":"15","author":"Ninomiya, Syoiti","year":"2008","journal-title":"Appl. Math. Finance","ISSN":"https:\/\/id.crossref.org\/issn\/1350-486X","issn-type":"print"},{"issue":"4","key":"32","first-page":"1635","article-title":"Stochastic Volterra equations with anticipating coefficients","volume":"18","author":"Pardoux, \u00c9tienne","year":"1990","journal-title":"Ann. Probab.","ISSN":"https:\/\/id.crossref.org\/issn\/0091-1798","issn-type":"print"},{"issue":"2","key":"33","first-page":"519","article-title":"Volterra equations driven by semimartingales","volume":"13","author":"Protter, Philip","year":"1985","journal-title":"Ann. Probab.","ISSN":"https:\/\/id.crossref.org\/issn\/0091-1798","issn-type":"print"},{"key":"34","doi-asserted-by":"publisher","first-page":"109","DOI":"10.1016\/j.spa.2021.07.003","article-title":"Discrete-time simulation of stochastic Volterra equations","volume":"141","author":"Richard, Alexandre","year":"2021","journal-title":"Stochastic Process. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0304-4149","issn-type":"print"},{"issue":"1","key":"35","doi-asserted-by":"publisher","first-page":"223","DOI":"10.1137\/21M1443807","article-title":"On the discrete-time simulation of the rough Heston model","volume":"14","author":"Richard, Alexandre","year":"2023","journal-title":"SIAM J. Financial Math."},{"key":"36","unstructured":"S. E. R\u00f8mer, Hybrid multifactor scheme for stochastic Volterra equations with completely monotone kernels, preprint, 2022, DOI 10.2139\/ssrn.3706253."},{"issue":"1","key":"37","doi-asserted-by":"publisher","first-page":"901","DOI":"10.1137\/16M1067986","article-title":"Construction of a third-order K-scheme and its application to financial models","volume":"8","author":"Shinozaki, Yuji","year":"2017","journal-title":"SIAM J. Financial Math."},{"issue":"4","key":"38","doi-asserted-by":"publisher","first-page":"483","DOI":"10.1080\/07362999008809220","article-title":"Expansion of the global error for numerical schemes solving stochastic differential equations","volume":"8","author":"Talay, Denis","year":"1990","journal-title":"Stochastic Anal. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0736-2994","issn-type":"print"},{"issue":"9","key":"39","doi-asserted-by":"publisher","first-page":"1062","DOI":"10.1016\/j.spl.2007.10.007","article-title":"Existence and uniqueness of solutions to stochastic Volterra equations with singular kernels and non-Lipschitz coefficients","volume":"78","author":"Wang, Zhidong","year":"2008","journal-title":"Statist. Probab. Lett.","ISSN":"https:\/\/id.crossref.org\/issn\/0167-7152","issn-type":"print"},{"key":"40","series-title":"Princeton Mathematical Series, vol. 6","volume-title":"The Laplace Transform","author":"Widder, David Vernon","year":"1941"},{"issue":"9","key":"41","doi-asserted-by":"publisher","first-page":"2226","DOI":"10.1016\/j.jde.2008.02.019","article-title":"Euler schemes and large deviations for stochastic Volterra equations with singular kernels","volume":"244","author":"Zhang, Xicheng","year":"2008","journal-title":"J. Differential Equations","ISSN":"https:\/\/id.crossref.org\/issn\/0022-0396","issn-type":"print"},{"issue":"4","key":"42","doi-asserted-by":"publisher","first-page":"1361","DOI":"10.1016\/j.jfa.2009.11.006","article-title":"Stochastic Volterra equations in Banach spaces and stochastic partial differential equation","volume":"258","author":"Zhang, Xicheng","year":"2010","journal-title":"J. Funct. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0022-1236","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.ams.org\/mcom\/2024-93-346\/S0025-5718-2023-03911-4\/mcom3911_AM.pdf","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https:\/\/www.ams.org\/mcom\/2024-93-346\/S0025-5718-2023-03911-4\/S0025-5718-2023-03911-4.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T05:12:11Z","timestamp":1776834731000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2024-93-346\/S0025-5718-2023-03911-4\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,11,2]]},"references-count":42,"journal-issue":{"issue":"346","published-print":{"date-parts":[[2024,3]]}},"alternative-id":["S0025-5718-2023-03911-4"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/3911","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":[[2023,11,2]]}}}