{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,2]],"date-time":"2026-01-02T07:51:45Z","timestamp":1767340305878,"version":"3.27.0"},"reference-count":44,"publisher":"Walter de Gruyter GmbH","issue":"3","funder":[{"DOI":"10.13039\/501100002241","name":"Japan Science and Technology Agency","doi-asserted-by":"publisher","award":["JPMJPR2029"],"award-info":[{"award-number":["JPMJPR2029"]}],"id":[{"id":"10.13039\/501100002241","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2023,9,1]]},"abstract":"Abstract<\/jats:title>The paper gives new results for the Milstein scheme of stochastic differential equations. We show that (i) the Milstein scheme holds as a weak approximation in total variation sense and is given by second-order polynomials of Brownian motion without using iterated integrals under non-commutative vector fields; (ii) the accuracy of the Milstein scheme is better than that of the Euler\u2013Maruyama scheme in an asymptotic sense. In particular, we proved<\/m:mi>TV<\/m:mi><\/m:msub>\u2062<\/m:mo>(<\/m:mo>X<\/m:mi>T<\/m:mi>\u03b5<\/m:mi><\/m:msubsup>,<\/m:mo>X<\/m:mi>\u00af<\/m:mo><\/m:mover>T<\/m:mi>\u03b5<\/m:mi>,<\/m:mo>Mil<\/m:mi>,<\/m:mo>(<\/m:mo>n<\/m:mi>)<\/m:mo><\/m:mrow><\/m:mrow><\/m:msubsup>)<\/m:mo><\/m:mrow><\/m:mrow>\u2264<\/m:mo>C<\/m:mi>\u2062<\/m:mo>\u03b5<\/m:mi>2<\/m:mn><\/m:msup><\/m:mrow>\/<\/m:mo>n<\/m:mi><\/m:mrow><\/m:mrow><\/m:math>d_{\\mathrm{TV}}(X_{T}^{\\varepsilon},\\bar{X}_{T}^{\\varepsilon,\\mathrm{Mil},(n)})\\leq C\\varepsilon^{2}\/n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>andd<\/m:mi>TV<\/m:mi><\/m:msub>\u2062<\/m:mo>(<\/m:mo>X<\/m:mi>T<\/m:mi>\u03b5<\/m:mi><\/m:msubsup>,<\/m:mo>X<\/m:mi>\u00af<\/m:mo><\/m:mover>T<\/m:mi>\u03b5<\/m:mi>,<\/m:mo>EM<\/m:mi>,<\/m:mo>(<\/m:mo>n<\/m:mi>)<\/m:mo><\/m:mrow><\/m:mrow><\/m:msubsup>)<\/m:mo><\/m:mrow><\/m:mrow>\u2264<\/m:mo>C<\/m:mi>\u2062<\/m:mo>\u03b5<\/m:mi><\/m:mrow>\/<\/m:mo>n<\/m:mi><\/m:mrow><\/m:mrow><\/m:math>d_{\\mathrm{TV}}(X_{T}^{\\varepsilon},\\bar{X}_{T}^{\\varepsilon,\\mathrm{EM},(n)})\\leq C\\varepsilon\/n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, whered<\/m:mi>TV<\/m:mi><\/m:msub><\/m:math>d_{\\mathrm{TV}}<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>is the total variation distance,X<\/m:mi>\u03b5<\/m:mi><\/m:msup><\/m:math>X^{\\varepsilon}<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>is a solution of a stochastic differential equation with a small parameter \ud835\udf00, andX<\/m:mi>\u00af<\/m:mo><\/m:mover>\u03b5<\/m:mi>,<\/m:mo>Mil<\/m:mi>,<\/m:mo>(<\/m:mo>n<\/m:mi>)<\/m:mo><\/m:mrow><\/m:mrow><\/m:msup><\/m:math>\\bar{X}^{\\varepsilon,\\mathrm{Mil},(n)}<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>andX<\/m:mi>\u00af<\/m:mo><\/m:mover>\u03b5<\/m:mi>,<\/m:mo>EM<\/m:mi>,<\/m:mo>(<\/m:mo>n<\/m:mi>)<\/m:mo><\/m:mrow><\/m:mrow><\/m:msup><\/m:math>\\bar{X}^{\\varepsilon,\\mathrm{EM},(n)}<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>are the Milstein scheme without iterated integrals and the Euler\u2013Maruyama scheme, respectively. In computational aspect, the scheme is useful to estimate probability distribution functions by a simple simulation without L\u00e9vy area computation. Numerical examples demonstrate the validity of the method.<\/jats:p>","DOI":"10.1515\/mcma-2023-2007","type":"journal-article","created":{"date-parts":[[2023,5,25]],"date-time":"2023-05-25T09:21:24Z","timestamp":1685006484000},"page":"221-242","source":"Crossref","is-referenced-by-count":1,"title":["Total variation bound for Milstein scheme without iterated integrals"],"prefix":"10.1515","volume":"29","author":[{"given":"Toshihiro","family":"Yamada","sequence":"first","affiliation":[{"name":"Hitotsubashi University , Tokyo , Japan"}]}],"member":"374","published-online":{"date-parts":[[2023,5,26]]},"reference":[{"key":"2023083109202740403_j_mcma-2023-2007_ref_001","doi-asserted-by":"crossref","unstructured":"C. J. S. Alves and A. B. Cruzeiro, Monte-Carlo simulation of stochastic differential systems\u2014a geometrical approach, Stochastic Process. Appl. 118 (2008), no. 3, 346\u2013367.","DOI":"10.1016\/j.spa.2007.04.009"},{"key":"2023083109202740403_j_mcma-2023-2007_ref_002","doi-asserted-by":"crossref","unstructured":"M. Arnaudon and A. Thalmaier, The differentiation of hypoelliptic diffusion semigroups, Illinois J. Math. 54 (2010), no. 4, 1285\u20131311.","DOI":"10.1215\/ijm\/1348505529"},{"key":"2023083109202740403_j_mcma-2023-2007_ref_003","doi-asserted-by":"crossref","unstructured":"V. Bally, L. Caramellino and R. Cont, Stochastic Integration by Parts and Functional It\u00f4 Calculus, Adv. Courses Math. CRM Barcelona, Birkh\u00e4user\/Springer, Cham, 2016.","DOI":"10.1007\/978-3-319-27128-6"},{"key":"2023083109202740403_j_mcma-2023-2007_ref_004","doi-asserted-by":"crossref","unstructured":"V. Bally and D. Talay, The law of the Euler scheme for stochastic differential equations. I. Convergence rate of the distribution function, Probab. Theory Related Fields 104 (1996), no. 1, 43\u201360.","DOI":"10.1007\/BF01303802"},{"key":"2023083109202740403_j_mcma-2023-2007_ref_005","doi-asserted-by":"crossref","unstructured":"V. Bally and D. Talay, The law of the Euler scheme for stochastic differential equations. II. Convergence rate of the density, Monte Carlo Methods Appl. 2 (1996), no. 2, 93\u2013128.","DOI":"10.1515\/mcma.1996.2.2.93"},{"key":"2023083109202740403_j_mcma-2023-2007_ref_006","doi-asserted-by":"crossref","unstructured":"C. Bayer, P. Friz and R. Loeffen, Semi-closed form cubature and applications to financial diffusion models, Quant. Finance 13 (2013), no. 5, 769\u2013782.","DOI":"10.1080\/14697688.2012.752102"},{"key":"2023083109202740403_j_mcma-2023-2007_ref_007","doi-asserted-by":"crossref","unstructured":"D. Crisan, K. Manolarakis and C. Nee, Cubature methods and applications, Paris-Princeton Lectures on Mathematical Finance 2013, Lecture Notes in Math. 2081, Springer, Cham (2013), 203\u2013316.","DOI":"10.1007\/978-3-319-00413-6_4"},{"key":"2023083109202740403_j_mcma-2023-2007_ref_008","doi-asserted-by":"crossref","unstructured":"A. B. Cruzeiro, P. Malliavin and A. Thalmaier, Geometrization of Monte-Carlo numerical analysis of an elliptic operator: Strong approximation, C. R. Math. Acad. Sci. Paris 338 (2004), no. 6, 481\u2013486.","DOI":"10.1016\/j.crma.2004.01.007"},{"key":"2023083109202740403_j_mcma-2023-2007_ref_009","unstructured":"A. M. Davie, Differential equations driven by rough paths: An approach via discrete approximation, Appl. Math. Res. Express. AMRX 2008 (2008), Article ID abm009."},{"key":"2023083109202740403_j_mcma-2023-2007_ref_010","doi-asserted-by":"crossref","unstructured":"A. M. Davie, KMT theory applied to approximations of SDE, Stochastic Analysis and Applications 2014, Springer Proc. Math. Stat. 100, Springer, Cham (2014), 185\u2013201.","DOI":"10.1007\/978-3-319-11292-3_7"},{"key":"2023083109202740403_j_mcma-2023-2007_ref_011","unstructured":"A. M. Davie, Pathwise approximation of stochastic differential equations using coupling, unpublished paper (2015), https:\/\/www.maths.ed.ac.uk\/adavie\/coum.pdf."},{"key":"2023083109202740403_j_mcma-2023-2007_ref_012","doi-asserted-by":"crossref","unstructured":"P. Del Moral, Mean Field Simulation for Monte Carlo Integration, Monogr. Statist. Appl. Probab. 126, CRC Press, Boca Raton, 2013.","DOI":"10.1201\/b14924"},{"key":"2023083109202740403_j_mcma-2023-2007_ref_013","doi-asserted-by":"crossref","unstructured":"P. Del Moral, J. Jacod and P. Protter, The Monte-Carlo method for filtering with discrete-time observations, Probab. Theory Related Fields 120 (2001), no. 3, 346\u2013368.","DOI":"10.1007\/PL00008786"},{"key":"2023083109202740403_j_mcma-2023-2007_ref_014","doi-asserted-by":"crossref","unstructured":"P. K. Friz and M. Hairer, A Course on Rough Paths, Universitext, Springer, Cham, 2014.","DOI":"10.1007\/978-3-319-08332-2"},{"key":"2023083109202740403_j_mcma-2023-2007_ref_015","doi-asserted-by":"crossref","unstructured":"P. K. Friz and N. B. Victoir, Multidimensional Stochastic Processes as Rough Paths: Theory and Applications, Cambridge Stud. Adv. Math. 120, Cambridge University, Cambridge, 2010.","DOI":"10.1017\/CBO9780511845079"},{"key":"2023083109202740403_j_mcma-2023-2007_ref_016","doi-asserted-by":"crossref","unstructured":"C. Graham and D. Talay, Stochastic Simulation and Monte Carlo Methods, Stoch. Model. Appl. Probab. 68, Springer, Heidelberg, 2013.","DOI":"10.1007\/978-3-642-39363-1"},{"key":"2023083109202740403_j_mcma-2023-2007_ref_017","doi-asserted-by":"crossref","unstructured":"L. G. Gyurk\u00f3 and T. J. Lyons, Efficient and practical implementations of cubature on Wiener space, Stochastic Analysis 2010, Springer, Heidelberg (2011), 73\u2013111.","DOI":"10.1007\/978-3-642-15358-7_5"},{"key":"2023083109202740403_j_mcma-2023-2007_ref_018","doi-asserted-by":"crossref","unstructured":"S. Hayakawa and K. Tanaka, Monte Carlo construction of cubature on Wiener space, Jpn. J. Ind. Appl. Math. 39 (2022), no. 2, 543\u2013571.","DOI":"10.1007\/s13160-021-00496-6"},{"key":"2023083109202740403_j_mcma-2023-2007_ref_019","doi-asserted-by":"crossref","unstructured":"Y. Hu and S. Watanabe, Donsker\u2019s delta functions and approximation of heat kernels by the time discretization methods, J. Math. Kyoto Univ. 36 (1996), no. 3, 499\u2013518.","DOI":"10.1215\/kjm\/1250518506"},{"key":"2023083109202740403_j_mcma-2023-2007_ref_020","unstructured":"N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, 2nd ed., North-Holland Math. Libr. 24, North-Holland, Amsterdam, 1989."},{"key":"2023083109202740403_j_mcma-2023-2007_ref_021","doi-asserted-by":"crossref","unstructured":"P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Appl. Math. (New York) 23, Springer, Berlin, 1992.","DOI":"10.1007\/978-3-662-12616-5"},{"key":"2023083109202740403_j_mcma-2023-2007_ref_022","doi-asserted-by":"crossref","unstructured":"S. Kusuoka, Approximation of expectation of diffusion process and mathematical finance, Taniguchi Conference on Mathematics Nara \u201998, Adv. Stud. Pure Math. 31, Mathematical Society of Japan, Tokyo (2001), 147\u2013165.","DOI":"10.2969\/aspm\/03110147"},{"key":"2023083109202740403_j_mcma-2023-2007_ref_023","doi-asserted-by":"crossref","unstructured":"S. Kusuoka, Approximation of expectation of diffusion processes based on Lie algebra and Malliavin calculus, Advances in Mathematical Economics. Vol. 6, Adv. Math. Econ. 6, Springer, Tokyo (2004), 69\u201383.","DOI":"10.1007\/978-4-431-68450-3_4"},{"key":"2023083109202740403_j_mcma-2023-2007_ref_024","doi-asserted-by":"crossref","unstructured":"S. Kusuoka and D. Stroock, Applications of the Malliavin calculus. I, Stochastic Analysis (Katata\/Kyoto 1982), North-Holland Math. Library 32, North-Holland, Amsterdam (1984), 271\u2013306.","DOI":"10.1016\/S0924-6509(08)70397-0"},{"key":"2023083109202740403_j_mcma-2023-2007_ref_025","unstructured":"S. Kusuoka and D. Stroock, Applications of the Malliavin calculus. II, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 32 (1985), no. 1, 1\u201376."},{"key":"2023083109202740403_j_mcma-2023-2007_ref_026","doi-asserted-by":"crossref","unstructured":"C. Litterer and T. Lyons, High order recombination and an application to cubature on Wiener space, Ann. Appl. Probab. 22 (2012), no. 4, 1301\u20131327.","DOI":"10.1214\/11-AAP786"},{"key":"2023083109202740403_j_mcma-2023-2007_ref_027","doi-asserted-by":"crossref","unstructured":"T. Lyons and N. Victoir, Cubature on Wiener space, Proc. Roy. Soc. Lond. A 460 (2004), 169\u2013198.","DOI":"10.1098\/rspa.2003.1239"},{"key":"2023083109202740403_j_mcma-2023-2007_ref_028","unstructured":"P. Malliavin and A. Thalmaier, Stochastic Calculus of Variations in Mathematical Finance, Springer Finance, Springer, Berlin, 2006."},{"key":"2023083109202740403_j_mcma-2023-2007_ref_029","doi-asserted-by":"crossref","unstructured":"G. Maruyama, Continuous Markov processes and stochastic equations, Rend. Circ. Mat. Palermo (2) 4 (1955), 48\u201390.","DOI":"10.1007\/BF02846028"},{"key":"2023083109202740403_j_mcma-2023-2007_ref_030","doi-asserted-by":"crossref","unstructured":"G. N. Mil\u0161te\u012dn, Approximate integration of stochastic differential equations, Theory Probab. Appl. 19 (1974), 557\u2013562.","DOI":"10.1137\/1119062"},{"key":"2023083109202740403_j_mcma-2023-2007_ref_031","doi-asserted-by":"crossref","unstructured":"S. Ninomiya and N. Victoir, Weak approximation of stochastic differential equations and application to derivative pricing, Appl. Math. Finance 15 (2008), no. 1\u20132, 107\u2013121.","DOI":"10.1080\/13504860701413958"},{"key":"2023083109202740403_j_mcma-2023-2007_ref_032","unstructured":"D. Nualart, The Malliavin Calculus and Related Topics, Probab. Appl. (New York), Springer, Berlin, 2006."},{"key":"2023083109202740403_j_mcma-2023-2007_ref_033","doi-asserted-by":"crossref","unstructured":"A. Takahashi and T. Yamada, An asymptotic expansion with push-down of Malliavin weights, SIAM J. Financial Math. 3 (2012), no. 1, 95\u2013136.","DOI":"10.1137\/100807624"},{"key":"2023083109202740403_j_mcma-2023-2007_ref_034","doi-asserted-by":"crossref","unstructured":"A. Takahashi and T. Yamada, On error estimates for asymptotic expansions with Malliavin weights: Application to stochastic volatility model, Math. Oper. Res. 40 (2015), no. 3, 513\u2013541.","DOI":"10.1287\/moor.2014.0683"},{"key":"2023083109202740403_j_mcma-2023-2007_ref_035","doi-asserted-by":"crossref","unstructured":"A. Takahashi and T. Yamada, A weak approximation with asymptotic expansion and multidimensional Malliavin weights, Ann. Appl. Probab. 26 (2016), no. 2, 818\u2013856.","DOI":"10.1214\/15-AAP1105"},{"key":"2023083109202740403_j_mcma-2023-2007_ref_036","unstructured":"S. Takanobu and S. Watanabe, Asymptotic expansion formulas of the Schilder type for a class of conditional Wiener functional integrations, Asymptotic Problems in Probability Theory: Wiener Functionals and Asymptotics (Sanda\/Kyoto 1990), Pitman Res. Notes Math. Ser. 284, Longman Scientific & Technical, Harlow (1993), 194\u2013241."},{"key":"2023083109202740403_j_mcma-2023-2007_ref_037","unstructured":"S. Watanabe, Lectures on Stochastic Differential Equations and Malliavin Calculus, Tata Inst. Fundam. Res. Lectures Math. Phys. 73, Springer, Berlin, 1984."},{"key":"2023083109202740403_j_mcma-2023-2007_ref_038","doi-asserted-by":"crossref","unstructured":"S. Watanabe, Analysis of Wiener functionals (Malliavin calculus) and its applications to heat kernels, Ann. Probab. 15 (1987), no. 1, 1\u201339.","DOI":"10.1214\/aop\/1176992255"},{"key":"2023083109202740403_j_mcma-2023-2007_ref_039","doi-asserted-by":"crossref","unstructured":"T. Yamada, Weak Milstein scheme without commutativity condition and its error bound, Appl. Numer. Math. 131 (2018), 95\u2013108.","DOI":"10.1016\/j.apnum.2018.04.007"},{"key":"2023083109202740403_j_mcma-2023-2007_ref_040","doi-asserted-by":"crossref","unstructured":"T. Yamada, An arbitrary high order weak approximation of SDE and Malliavin Monte Carlo: Analysis of probability distribution functions, SIAM J. Numer. Anal. 57 (2019), no. 2, 563\u2013591.","DOI":"10.1137\/17M114412X"},{"key":"2023083109202740403_j_mcma-2023-2007_ref_041","doi-asserted-by":"crossref","unstructured":"T. Yamada, High order weak approximation for irregular functionals of time-inhomogeneous SDEs, Monte Carlo Methods Appl. 27 (2021), no. 2, 117\u2013136.","DOI":"10.1515\/mcma-2021-2085"},{"key":"2023083109202740403_j_mcma-2023-2007_ref_042","doi-asserted-by":"crossref","unstructured":"T. Yamada, Short communication: A Gaussian Kusuoka approximation without solving random ODEs, SIAM J. Financial Math. 13 (2022), no. 1, SC1\u2013SC11.","DOI":"10.1137\/21M1433915"},{"key":"2023083109202740403_j_mcma-2023-2007_ref_043","doi-asserted-by":"crossref","unstructured":"T. Yamada and K. Yamamoto, A second-order discretization with Malliavin weight and quasi-Monte Carlo method for option pricing, Quant. Finance 20 (2020), no. 11, 1825\u20131837.","DOI":"10.1080\/14697688.2018.1430371"},{"key":"2023083109202740403_j_mcma-2023-2007_ref_044","doi-asserted-by":"crossref","unstructured":"N. Yoshida, Asymptotic expansions of maximum likelihood estimators for small diffusions via the theory of Malliavin\u2013Watanabe, Probab. Theory Related Fields 92 (1992), no. 3, 275\u2013311.","DOI":"10.1007\/BF01300558"}],"container-title":["Monte Carlo Methods and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/mcma-2023-2007\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/mcma-2023-2007\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,10,21]],"date-time":"2024-10-21T02:45:14Z","timestamp":1729478714000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/mcma-2023-2007\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,5,26]]},"references-count":44,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2023,7,4]]},"published-print":{"date-parts":[[2023,9,1]]}},"alternative-id":["10.1515\/mcma-2023-2007"],"URL":"https:\/\/doi.org\/10.1515\/mcma-2023-2007","relation":{},"ISSN":["0929-9629","1569-3961"],"issn-type":[{"type":"print","value":"0929-9629"},{"type":"electronic","value":"1569-3961"}],"subject":[],"published":{"date-parts":[[2023,5,26]]}}}