{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:43:29Z","timestamp":1787323409308,"version":"3.56.0"},"reference-count":21,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1615229"],"award-info":[{"award-number":["DMS-1615229"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-2008427"],"award-info":[{"award-number":["DMS-2008427"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-2038118"],"award-info":[{"award-number":["DMS-2038118"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Finan. Math."],"published-print":{"date-parts":[[2025,12,31]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>The strong relative arbitrage problem in stochastic portfolio theory seeks an investment strategy that almost surely outperforms a benchmark portfolio at the end of a given time horizon. The highest relative return in relative arbitrage opportunities is characterized by the smallest nonnegative continuous solution of a Cauchy problem for a partial differential equation (PDE) [D. Fernholz and I. Karatzas, Ann. Appl. Probab., 20 (2010), pp. 1179\u20131204]. However, solving this type of PDE poses analytical and numerical challenges, due to the high dimensionality and multiple solutions of the PDEs. In this paper, we develop a numerical method to solve the relative arbitrage problem and the associated PDE in a volatility-stabilized market model, formulating the capitalization as time-changed Bessel bridges, and present a practical algorithm and demonstrate numerical results for optimal arbitrage opportunities.<\/jats:p>","DOI":"10.1137\/25m1729216","type":"journal-article","created":{"date-parts":[[2025,10,14]],"date-time":"2025-10-14T14:34:54Z","timestamp":1760452494000},"page":"SC76-SC87","source":"Crossref","is-referenced-by-count":2,"title":["Short Communication: Finding the Nonnegative Minimal Solutions of Cauchy PDEs in a Volatility-Stabilized Market"],"prefix":"10.1137","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6699-9150","authenticated-orcid":true,"given":"Nicole Tianjiao","family":"Yang","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Tennessee, Knoxville, TN 37916 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2787-7569","authenticated-orcid":true,"given":"Tomoyuki","family":"Ichiba","sequence":"additional","affiliation":[{"name":"Department of Statistics and Applied Probability, University of California, Santa Barbara, CA 93106 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2025,10,14]]},"reference":[{"key":"ref1","unstructured":"Y. Achdou and Z. Kobeissi, Mean Field Games of Controls: Finite Difference Approximations, preprint, arXiv:2003.03968, 2020."},{"key":"ref2","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-10-10306-2"},{"key":"ref3","doi-asserted-by":"publisher","DOI":"10.1090\/btran\/141"},{"key":"ref4","doi-asserted-by":"publisher","DOI":"10.1214\/23-ECP524"},{"key":"ref5","doi-asserted-by":"publisher","DOI":"10.1080\/14697688.2010.495078"},{"key":"ref6","doi-asserted-by":"publisher","DOI":"10.1214\/08-AAP579"},{"key":"ref7","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2010.04.014"},{"key":"ref8","doi-asserted-by":"publisher","DOI":"10.1214\/aop\/1024404416"},{"key":"ref9","doi-asserted-by":"publisher","DOI":"10.1214\/09-AAP642"},{"key":"ref10","doi-asserted-by":"publisher","DOI":"10.1214\/10-AAP755"},{"key":"ref11","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-3699-1"},{"key":"ref12","doi-asserted-by":"publisher","DOI":"10.1007\/s10436-004-0011-6"},{"key":"ref13","volume-title":"Bessel and Volatility-Stabilized Processes","author":"Goia I.","year":"2009"},{"key":"ref14","doi-asserted-by":"publisher","DOI":"10.1090\/mcom\/3514"},{"key":"ref15","doi-asserted-by":"publisher","DOI":"10.1214\/10-AAP706"},{"key":"ref16","unstructured":"T. Ichiba and T. Yang, Relative Arbitrage Opportunities with Interactions Among \\(N\\) Investors, preprint, arXiv:2006.15158, 2020."},{"key":"ref17","volume-title":"Stochastic Differential Equations: An Introduction with Applications","author":"Oksendal B.","year":"2013"},{"key":"ref18","doi-asserted-by":"publisher","DOI":"10.1214\/10-AAP725"},{"key":"ref19","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-06400-9"},{"key":"ref20","unstructured":"N. T. Yang and T. Ichiba, Relative Arbitrage Opportunities in an Extended Mean Field System, preprint, arXiv:2311.02690, 2023."},{"key":"ref21","volume-title":"Topics in Relative Arbitrage, Stochastic Games and High-Dimensional PDEs","author":"Yang T.","year":"2021"}],"container-title":["SIAM Journal on Financial Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/25M1729216","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:00:58Z","timestamp":1787320858000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/25M1729216"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,10,14]]},"references-count":21,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2025,12,31]]}},"alternative-id":["10.1137\/25M1729216"],"URL":"https:\/\/doi.org\/10.1137\/25m1729216","relation":{},"ISSN":["1945-497X"],"issn-type":[{"value":"1945-497X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,10,14]]}}}