{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,13]],"date-time":"2025-11-13T07:17:29Z","timestamp":1763018249228,"version":"build-2065373602"},"reference-count":25,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2020,8,1]],"date-time":"2020-08-01T00:00:00Z","timestamp":1596240000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001871","name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","doi-asserted-by":"publisher","award":["UID\/MAT\/04106\/2019 (CIDMA)"],"award-info":[{"award-number":["UID\/MAT\/04106\/2019 (CIDMA)"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Fractal Fract"],"abstract":"<jats:p>We introduce a stochastic fractional calculus. As an application, we present a stochastic fractional calculus of variations, which generalizes the fractional calculus of variations to stochastic processes. A stochastic fractional Euler\u2013Lagrange equation is obtained, extending those available in the literature for the classical, fractional, and stochastic calculus of variations. To illustrate our main theoretical result, we discuss two examples: one derived from quantum mechanics, the second validated by an adequate numerical simulation.<\/jats:p>","DOI":"10.3390\/fractalfract4030038","type":"journal-article","created":{"date-parts":[[2020,8,3]],"date-time":"2020-08-03T03:16:46Z","timestamp":1596424606000},"page":"38","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["A Stochastic Fractional Calculus with Applications to Variational Principles"],"prefix":"10.3390","volume":"4","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8412-7783","authenticated-orcid":false,"given":"Houssine","family":"Zine","sequence":"first","affiliation":[{"name":"Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8641-2505","authenticated-orcid":false,"given":"Delfim F. M.","family":"Torres","sequence":"additional","affiliation":[{"name":"Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2020,8,1]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1016\/0022-1236(81)90079-3","article-title":"Stochastic calculus of variations","volume":"41","author":"Yasue","year":"1981","journal-title":"J. Funct. Anal."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"168","DOI":"10.1016\/j.aml.2018.05.005","article-title":"A stochastic SICA epidemic model for HIV transmission","volume":"84","author":"Djordjevic","year":"2018","journal-title":"Appl. Math. 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