{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:43:55Z","timestamp":1787323435220,"version":"3.56.0"},"reference-count":42,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"5","funder":[{"DOI":"10.13039\/100000879","name":"Alfred P. Sloan Foundation","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100000879","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund","doi-asserted-by":"publisher","award":["Y782"],"award-info":[{"award-number":["Y782"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100011102","name":"Seventh Framework Programme","doi-asserted-by":"publisher","award":["335421"],"award-info":[{"award-number":["335421"]}],"id":[{"id":"10.13039\/100011102","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000038","name":"Natural Sciences and Engineering Research Council of Canada","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100000038","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012600","name":"ShanghaiTech University","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100012600","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100005247","name":"University of British Columbia","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100005247","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Control Optim."],"published-print":{"date-parts":[[2020,1]]},"abstract":"<jats:p>Given two probability measures $\\mu, \\nu$ on $\\mathbb{R}^d$, in subharmonic order, we describe optimal stopping times $\\tau$ that maximize\/minimize the cost functional $\\mathbb{E} |B_0 - B_\\tau|^{\\alpha}$, $\\alpha &gt; 0$, where $(B_t)_t$ is Brownian motion with initial law $\\mu$ and with final distribution---once stopped at $\\tau$---equal to $\\nu$. Under the assumption of radial symmetry on $\\mu$ and $\\nu$, we show that in dimension $d \\geq 3$ and $\\alpha \\neq 2$, there exists a unique optimal solution given by a nonrandomized stopping time characterized as the hitting time to a suitably symmetric barrier. We also relate this problem to the optimal transportation problem for subharmonic martingales and establish a duality result.<\/jats:p>","DOI":"10.1137\/19m1270513","type":"journal-article","created":{"date-parts":[[2020,9,15]],"date-time":"2020-09-15T11:10:21Z","timestamp":1600168221000},"page":"2765-2789","source":"Crossref","is-referenced-by-count":7,"title":["Optimal Brownian Stopping When the Source and Target Are Radially Symmetric Distributions"],"prefix":"10.1137","volume":"58","author":[{"given":"Nassif","family":"Ghoussoub","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6920-603X","authenticated-orcid":true,"given":"Young-Heon","family":"Kim","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1290-6964","authenticated-orcid":true,"given":"Tongseok","family":"Lim","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2020,9,15]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1215\/ijm\/1256051015"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1007\/s00222-016-0692-2"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1007\/s00780-013-0205-8"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1016\/j.spa.2017.01.004"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1214\/14-AOP966"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1214\/16-AOP1131"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.3160440402"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1992-1035999-6"},{"key":"atypb9","first-page":"19","author":"Chacon R. V.","year":"1976","journal-title":"Berlin"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.5802\/aif.87"},{"key":"atypb11","first-page":"173","author":"Cox A. M. G.","year":"2018","journal-title":"Probab. Theory Relat. Fields"},{"key":"atypb12","unstructured":"H. De March and N. Touzi,\n                      Irreducible Convex Paving for Decomposition of Multi-Dimensional Martingale Transport Plans\n                      ,https:\/\/arxiv.org\/abs\/1702.08298, 2017."},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1007\/s00440-013-0531-y"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1016\/j.spa.2015.05.009"},{"key":"atypb15","first-page":"38","author":"Edgar G. A.","year":"1984","journal-title":"New York"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1016\/0022-1236(86)90092-3"},{"key":"atypb17","first-page":"357","author":"Falkner N.","year":"1980","journal-title":"Berlin"},{"key":"atypb18","unstructured":"G. B. Folland,\n                      Real Analysis: Modern Techniques and Their Applications\n                      , John Wiley & Sons, New York, 2013."},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1214\/13-AAP925"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1007\/BF02392620"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1214\/18-AOP1258"},{"key":"atypb22","unstructured":"N. Ghoussoub, Y.H. Kim, and A. Z. Palmer,\n                      A Solution to the Monge Transport Problem for Brownian Martingales\n                      ,arXiv:1903.00527, 2019."},{"key":"atypb23","first-page":"1","volume":"58","author":"Ghoussoub N.","journal-title":"Calc. Var. Partial Differential Equations"},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.5802\/aif.1198"},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1137\/15M1025256"},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1007\/s007800050044"},{"key":"atypb27","first-page":"267","author":"Hobson D.","year":"2011","journal-title":"Berlin"},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1007\/s00780-014-0249-4"},{"key":"atypb29","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-9965.2010.00473.x"},{"key":"atypb30","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9904-1969-12417-1"},{"key":"atypb31","doi-asserted-by":"publisher","DOI":"10.1214\/16-AAP1191"},{"key":"atypb32","doi-asserted-by":"publisher","DOI":"10.2307\/1971328"},{"key":"atypb33","unstructured":"T. Lim,\n                      Optimal Martingale Transport between Radially Symmetric Marginals in General Dimensions\n                      ,https:\/\/arxiv.org\/abs\/1412.3530, 2016."},{"key":"atypb34","doi-asserted-by":"publisher","DOI":"10.1214\/aoms\/1177692480"},{"key":"atypb35","doi-asserted-by":"publisher","DOI":"10.1214\/154957804100000060"},{"key":"atypb36","unstructured":"J. Ob\u0142\u00f3j and P. Siorpaes,\n                      Structure of Martingale Transports in Finite Dimensions\n                      ,https:\/\/arxiv.org\/abs\/1702.08433, 2017."},{"key":"atypb37","doi-asserted-by":"publisher","DOI":"10.1214\/16-AOP1110"},{"key":"atypb38","doi-asserted-by":"publisher","DOI":"10.1007\/BF01418740"},{"key":"atypb39","unstructured":"A. V. Skorokhod,\n                      Studies in the Theory of Random Processes\n                      , Addison-Wesley, Reading, MA, 1965."},{"key":"atypb40","doi-asserted-by":"publisher","DOI":"10.1214\/aoms\/1177700153"},{"key":"atypb41","doi-asserted-by":"crossref","unstructured":"C. Villani,\n                      Topics in Optimal Transportation\n                      , Grad. Stud. Math. 58, AMS, Providence, RI, 2003.","DOI":"10.1090\/gsm\/058"},{"key":"atypb42","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9904-1929-04753-0"}],"container-title":["SIAM Journal on Control and Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/19M1270513","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:02:14Z","timestamp":1787320934000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/19M1270513"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,1]]},"references-count":42,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2020,1]]}},"alternative-id":["10.1137\/19M1270513"],"URL":"https:\/\/doi.org\/10.1137\/19m1270513","relation":{},"ISSN":["0363-0129","1095-7138"],"issn-type":[{"value":"0363-0129","type":"print"},{"value":"1095-7138","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,1]]}}}