{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:21:36Z","timestamp":1787329296216,"version":"build-2736575974"},"reference-count":60,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"6","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Numer. Anal."],"published-print":{"date-parts":[[2008,1]]},"abstract":"<jats:p>We present numerical schemes for the strong solution of linear stochastic differential equations driven by an arbitrary number of Wiener processes. These schemes are based on the Neumann (stochastic Taylor) and Magnus expansions. First, we consider the case when the governing linear diffusion vector fields commute with each other, but not with the linear drift vector field. We prove that numerical methods based on the Magnus expansion are more accurate in the mean-square sense than corresponding stochastic Taylor integration schemes. Second, we derive the maximal rate of convergence for arbitrary multidimensional stochastic integrals approximated by their conditional expectations. Consequently, for general nonlinear stochastic differential equations with noncommuting vector fields, we deduce explicit formulae for the relation between error and computational costs for methods of arbitrary order. Third, we consider the consequences in two numerical studies, one of which is an application arising in stochastic linear-quadratic optimal control.<\/jats:p>","DOI":"10.1137\/060656486","type":"journal-article","created":{"date-parts":[[2008,8,6]],"date-time":"2008-08-06T18:07:40Z","timestamp":1218046060000},"page":"2892-2919","source":"Crossref","is-referenced-by-count":19,"title":["Efficient Strong Integrators for Linear Stochastic Systems"],"prefix":"10.1137","volume":"46","author":[{"given":"Gabriel","family":"Lord","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Simon J. A.","family":"Malham","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Anke","family":"Wiese","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2008,8,6]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1007\/s10543-005-0001-8"},{"key":"R2","unstructured":"L. Arnold,\n                      Stochastic Differential Equations\n                      , John Wiley & Sons, New York, 1974."},{"key":"R3","doi-asserted-by":"crossref","unstructured":"R. Azencott,\n                      Formule de Taylor stochastique et d\u00e9veloppement asymptotique d'int\u00e9grales de Feynman\n                      , Seminar on Probability XVI, Lecture Notes in Math., 921, Springer, Berlin, 1982, pp. 237\u2013285.","DOI":"10.1007\/BFb0092653"},{"key":"R4","doi-asserted-by":"crossref","unstructured":"F. Baudoin,\n                      An Introduction to the Geometry of Stochastic flows\n                      , Imperial College Press, London, 2004.","DOI":"10.1142\/p347"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1007\/BF00343737"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.2004.824468"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1016\/S0167-2789(99)00097-4"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1098\/rspa.2003.1247"},{"key":"R9","unstructured":"P. M. Burrage,\n                      Runge\u2013Kutta methods for stochastic differential equations\n                      , Ph.D. thesis, University of Queensland, Queensland, Australia, 1999."},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827500376922"},{"key":"R11","unstructured":"F. Casas and A. 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Skorohod,\n                      The Theory of Stochastic Processes\n                      III, Springer-Verlag, Berlin, 1979.","DOI":"10.1007\/978-1-4612-6202-2"},{"key":"R24","unstructured":"M. B. Giles,\n                      Multi-level Monte Carlo path simulation\n                      , OUCL, Report no. 06\/03, Oxford, UK."},{"key":"R25","unstructured":"M. B. Giles,\n                      Improved multilevel Monte Carlo convergence using the Milstein scheme\n                      , OUCL, Report no. 06\/22, Oxford, UK."},{"key":"R26","unstructured":"H. Gilsing and T. 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Milstein,\n                      Numerical Integration of Stochastic Differential Equations\n                      , Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, 1994."},{"key":"R45","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142901395588"},{"key":"R46","doi-asserted-by":"publisher","DOI":"10.1137\/S106482750037024X"},{"key":"R47","doi-asserted-by":"publisher","DOI":"10.1137\/S00361445024180"},{"key":"R48","doi-asserted-by":"publisher","DOI":"10.1007\/s10208-006-0222-5"},{"key":"R49","doi-asserted-by":"publisher","DOI":"10.1137\/0151028"},{"key":"R50","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142996307946"},{"key":"R51","unstructured":"H. Schurz,\n                      A brief introduction to numerical analysis of (ordinary) stochastic differential equations without tears\n                      , in Handbook of Stochastic Analysis and Applications, V. Lakshmikantham and D. 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