{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,30]],"date-time":"2026-09-30T08:53:43Z","timestamp":1790758423733,"version":"4.1.0"},"reference-count":28,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Finan. Math."],"published-print":{"date-parts":[[2011,1]]},"abstract":"<jats:p>We analyze the robustness properties of the Snell envelope backward evolution equation for the discrete time optimal stopping problem. We consider a series of approximation schemes, including cut\u2013off-type approximations, Euler discretization schemes, interpolation models, quantization tree models, and the stochastic mesh method of Broadie and Glasserman. In each situation, we provide nonasymptotic convergence estimates, including $\\mathbb{L}_p$-mean error bounds and exponential concentration inequalities. We deduce these estimates from a single and general robustness property of Snell envelope semigroups. In particular, this analysis allows us to recover existing convergence results for the quantization tree method and to improve significantly the rates of convergence obtained for the stochastic mesh estimator of Broadie and Glasserman. In the second part of the article, we propose a new approach based on a genealogical tree approximation model of the reference Markov process in terms of a neutral-type genetic model. In contrast to Broadie\u2013Glasserman Monte Carlo models, the computational cost of this new stochastic approximation is linear in the number of particles. Some simulation results are provided and confirm the interest of this new algorithm.<\/jats:p>","DOI":"10.1137\/100798016","type":"journal-article","created":{"date-parts":[[2011,8,25]],"date-time":"2011-08-25T18:32:23Z","timestamp":1314297143000},"page":"587-626","source":"Crossref","is-referenced-by-count":9,"title":["On the Robustness of the Snell Envelope"],"prefix":"10.1137","volume":"2","author":[{"given":"Pierre","family":"Del Moral","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Peng","family":"Hu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Nadia","family":"Oudjane","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Bruno","family":"R\u00e9millard","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2011,8,25]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.21314\/JCF.2004.118"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1111\/j.0960-1627.2005.00213.x"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1007\/BF01303802"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1515\/mcma.1996.2.2.93"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.21314\/JCF.2004.117"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1016\/j.spa.2004.01.001"},{"key":"R7","unstructured":"B. Bouchard and X. Warin,\n                      Monte-Carlo valorisation of American options: Facts and new algorithms to improve existing methods\n                      , in Numerical Methods in Finance, R. Carmona, P. Del Moral, P. Hu, and N. Oudjane, eds., Springer-Verlag, Berlin, to appear."},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.1982.1056486"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1214\/aoap\/1028903535"},{"key":"R10","doi-asserted-by":"crossref","unstructured":"P. Del Moral,\n                      Feynman-Kac Formula: Genealogical and Interacting Particle Systems with Applications\n                      , Prob. Appl. (N.Y.), Springer-Verlag, New York, 2004.","DOI":"10.1007\/978-1-4684-9393-1_3"},{"key":"R11","unstructured":"P. Del Moral, A. Doucet, and A. Jasra,\n                      On adaptive resampling procedures for sequential Monte Carlo methods\n                      , Bernoulli, to appear."},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1051\/m2an\/2010048"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1016\/S0246-0203(00)01064-5"},{"key":"R14","doi-asserted-by":"crossref","unstructured":"P. Del Moral, B. R\u00e9millard, and S. Rubenthaler,\n                      Monte Carlo approximations of American options that preserve monotonicity and convexity\n                      , SSRN eLibrary no. 1703906, 2010.","DOI":"10.2139\/ssrn.1703906"},{"key":"R15","unstructured":"P. Del Moral and E. Rio,\n                      Concentration inequalities for mean field particle models\n                      , Ann. Appl. Probab., to appear."},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1080\/07362990903415833"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1214\/105051605000000034"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1214\/105051605000000043"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1214\/105051607000000249"},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1214\/aoap\/1075828046"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1214\/105051605000000412"},{"key":"R22","doi-asserted-by":"crossref","unstructured":"S. Graf and H. Luschgy,\n                      Foundations of Quantization for Probability Distributions\n                      , Lecture Notes in Math. 1730, Springer-Verlag, Berlin, 2000.","DOI":"10.1007\/BFb0103945"},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1214\/aoap\/1075828050"},{"key":"R24","doi-asserted-by":"publisher","DOI":"10.1016\/j.orl.2009.06.001"},{"key":"R25","doi-asserted-by":"publisher","DOI":"10.1016\/S0377-0427(97)00190-8"},{"key":"R26","doi-asserted-by":"crossref","unstructured":"G. Pag\u00e8s, H. Pham, and J. Printems,\n                      Optimal quantization and applications to numerical problems in finance\n                      , in Handbook of Computational and Numerical Methods in Finance, S. T. Rachev, ed., Birkh\u00e4user, Boston, 2004, pp. 253\u2013297.","DOI":"10.1007\/978-0-8176-8180-7_7"},{"key":"R27","doi-asserted-by":"publisher","DOI":"10.1515\/156939605777438578"},{"key":"R28","doi-asserted-by":"publisher","DOI":"10.1016\/j.spa.2009.09.004"}],"container-title":["SIAM Journal on Financial Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/100798016","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:45:28Z","timestamp":1787319928000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/100798016"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,1]]},"references-count":28,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2011,1]]}},"alternative-id":["10.1137\/100798016"],"URL":"https:\/\/doi.org\/10.1137\/100798016","relation":{},"ISSN":["1945-497X"],"issn-type":[{"value":"1945-497X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2011,1]]}}}