{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,5]],"date-time":"2026-10-05T23:02:10Z","timestamp":1791241330925,"version":"4.1.0"},"reference-count":36,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"5","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Numer. Anal."],"published-print":{"date-parts":[[2008,1]]},"abstract":"<jats:p>A numerical method for a class of forward-backward stochastic differential equations (FBSDEs) is proposed and analyzed. The method is designed around the four step scheme [J. Douglas, Jr., J. Ma, and P. Protter, Ann. Appl. Probab., 6 (1996), pp. 940\u2013968] but with a Hermite-spectral method to approximate the solution to the decoupling quasi-linear PDE on the whole space. A rigorous synthetic error analysis is carried out for a fully discretized scheme, namely a first-order scheme in time and a Hermite-spectral scheme in space, of the FBSDEs. Equally important, a systematical numerical comparison is made between several schemes for the resulting decoupled forward SDE, including a stochastic version of the Adams\u2013Bashforth scheme. It is shown that the stochastic version of the Adams\u2013Bashforth scheme coupled with the Hermite-spectral method leads to a convergence rate of $\\frac 32$ (in time) which is better than those in previously published work for the FBSDEs.<\/jats:p>","DOI":"10.1137\/06067393x","type":"journal-article","created":{"date-parts":[[2008,7,10]],"date-time":"2008-07-10T14:11:44Z","timestamp":1215699104000},"page":"2636-2661","source":"Crossref","is-referenced-by-count":61,"title":["On Numerical Approximations of Forward-Backward Stochastic Differential Equations"],"prefix":"10.1137","volume":"46","author":[{"given":"Jin","family":"Ma","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jie","family":"Shen","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yanhong","family":"Zhao","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2008,7,2]]},"reference":[{"key":"R1","unstructured":"V. Bally,\n                      Approximation scheme for solutions of BSDE\n                      , in Backward Stochastic Differential Equations (Paris, 1995\u20131996), Pitman Res. Notes Math. Ser. 364, Longman, Harlow, UK, 1997, pp. 177\u2013191."},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.3150\/bj\/1072215199"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1111\/j.0960-1627.2005.00213.x"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1214\/07-AAP448"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1016\/j.spa.2004.01.001"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1214\/ECP.v6-1030"},{"key":"R7","doi-asserted-by":"crossref","unstructured":"H.J. Bungartz and T. Dornseifer,\n                      Sparse grids: Recent developments for elliptic partial differential equations\n                      , in Multigrid Methods V (Stuttgart, 1996), Lect. Notes Comput. Sci. 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A probabilistic scheme\n                      , in S\u00e9minaire de Probabilit\u00e9s XXXVII, Lecture Notes in Math. 1832, Springer, Berlin, 2003, pp. 290\u2013332.","DOI":"10.1007\/978-3-540-40004-2_12"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1214\/105051605000000674"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1214\/aoap\/1034968235"},{"key":"R14","unstructured":"N. El Karoui and L. Mazliak, eds.\n                      Backward Stochastic Differential Equations\n                      (Paris, 1995\u20131996), Pitman Res. Notes Math. Ser. 364, Longman, Harlow, UK, 1997."},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1111\/1467-9965.00022"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142902418333"},{"key":"R17","unstructured":"I. M. Gradshteyn and I. M. Ryzhik,\n                      Table of Integrals, Series, and Products\n                      , 6th ed., Academic Press, San Diego, CA, 2000."},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1023\/A:1022892132249"},{"key":"R19","doi-asserted-by":"crossref","unstructured":"F. J. Hickernell,\n                      Lattice rules: How well do they measure up?\n                      , in Random and Quasi-Random Point Sets, Lecture Notes in Statist. 138, Springer, New York, 1998, pp. 109\u2013166.","DOI":"10.1007\/978-1-4612-1702-2_3"},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.2969\/jmsj\/00310157"},{"key":"R21","doi-asserted-by":"crossref","unstructured":"P. E. Kloeden and E. Platen,\n                      Numerical Solution of Stochastic Differential Equations\n                      , Springer, Berlin, 1992.","DOI":"10.1007\/978-3-662-12616-5"},{"key":"R22","doi-asserted-by":"crossref","unstructured":"F. Y. Kuo, I. H. Sloan, and H. 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