{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,27]],"date-time":"2026-05-27T20:10:58Z","timestamp":1779912658864,"version":"3.53.1"},"reference-count":45,"publisher":"Elsevier BV","license":[{"start":{"date-parts":[[2026,9,1]],"date-time":"2026-09-01T00:00:00Z","timestamp":1788220800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/tdm\/userlicense\/1.0\/"},{"start":{"date-parts":[[2026,9,1]],"date-time":"2026-09-01T00:00:00Z","timestamp":1788220800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/legal\/tdmrep-license"},{"start":{"date-parts":[[2026,9,1]],"date-time":"2026-09-01T00:00:00Z","timestamp":1788220800000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-017"},{"start":{"date-parts":[[2026,9,1]],"date-time":"2026-09-01T00:00:00Z","timestamp":1788220800000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-037"},{"start":{"date-parts":[[2026,9,1]],"date-time":"2026-09-01T00:00:00Z","timestamp":1788220800000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-012"},{"start":{"date-parts":[[2026,9,1]],"date-time":"2026-09-01T00:00:00Z","timestamp":1788220800000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-029"},{"start":{"date-parts":[[2026,9,1]],"date-time":"2026-09-01T00:00:00Z","timestamp":1788220800000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-004"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["elsevier.com","sciencedirect.com"],"crossmark-restriction":true},"short-container-title":["Fuzzy Sets and Systems"],"published-print":{"date-parts":[[2026,9]]},"DOI":"10.1016\/j.fss.2026.109915","type":"journal-article","created":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T14:58:28Z","timestamp":1776697108000},"page":"109915","update-policy":"https:\/\/doi.org\/10.1016\/elsevier_cm_policy","source":"Crossref","is-referenced-by-count":0,"special_numbering":"C","title":["The It\u00f4-Henstock integral and its numerical calculation of the fuzzy stochastic process"],"prefix":"10.1016","volume":"538","author":[{"given":"Yaping","family":"Li","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5878-1506","authenticated-orcid":false,"given":"Zengtai","family":"Gong","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"78","reference":[{"issue":"8","key":"10.1016\/j.fss.2026.109915_bib0001","first-page":"519","article-title":"Stochastic integral","volume":"20","author":"It\u00f4","year":"1944","journal-title":"Proc. Imp. Acad. Tokyo"},{"key":"10.1016\/j.fss.2026.109915_bib0002","doi-asserted-by":"crossref","unstructured":"O. Kahouli, A. B. Makhlouf, L. Mchiri, H. Rguigui, Hyers\u2013Ulam stability for a class of Hadamard fractional It\u00f4\u2013Doob stochastic integral equations, Chaos Solitons Fractals 166(2023) 112918.","DOI":"10.1016\/j.chaos.2022.112918"},{"key":"10.1016\/j.fss.2026.109915_bib0003","doi-asserted-by":"crossref","unstructured":"M.H. Heydari, M.R. Hooshmandasl, C. Cattani, F.M. Maalek Ghaini, An efficient computational method for solving nonlinear stochastic It\u00f4-integral equations: application for stochastic problems in physics, J. Comput. Phys. 283(2015) 148\u2013168.","DOI":"10.1016\/j.jcp.2014.11.042"},{"key":"10.1016\/j.fss.2026.109915_bib0004","doi-asserted-by":"crossref","unstructured":"D. Cohen, J. Cui, J. Hong, L. Sun, Exponential integrators for stochastic Maxwell\u2019s equations driven by It\u00f4 noise, J. Comput. Phys. 410(2020) 109382.","DOI":"10.1016\/j.jcp.2020.109382"},{"key":"10.1016\/j.fss.2026.109915_bib0005","doi-asserted-by":"crossref","unstructured":"W. Li, B. Wang, D. Huang, V. Rajic, J. Zhao, Dynamical properties of a stochastic tumor-immune model with comprehensive pulsed therapy, Commun. Nonlinear Sci. Numer. Simul. 140 (2024) 108330.","DOI":"10.1016\/j.cnsns.2024.108330"},{"issue":"3","key":"10.1016\/j.fss.2026.109915_bib0006","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1007\/BF01258530","article-title":"Quantum Ito\u2019s formula and stochastic evolutions","volume":"93","author":"Hudson","year":"1984","journal-title":"Commun. Math. Phys."},{"issue":"7","key":"10.1016\/j.fss.2026.109915_bib0007","doi-asserted-by":"crossref","first-page":"4967","DOI":"10.1063\/1.533387","article-title":"Algebraic theory of product integrals in quantum stochastic calculus","volume":"41","author":"Hudson","year":"2000","journal-title":"J. Math. Phys."},{"key":"10.1016\/j.fss.2026.109915_bib0008","doi-asserted-by":"crossref","unstructured":"K.B. Sinha, Brownian bridge in quantum probability, Indian J. Pure Appl. Math. 48(2017) 551\u2013560.","DOI":"10.1007\/s13226-017-0245-4"},{"issue":"2","key":"10.1016\/j.fss.2026.109915_bib0009","doi-asserted-by":"crossref","first-page":"401","DOI":"10.1081\/SAP-120019292","article-title":"On set-valued stochastic integrals","volume":"21","author":"Jung","year":"2003","journal-title":"Stoch. Anal. Appl."},{"issue":"1","key":"10.1016\/j.fss.2026.109915_bib0010","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0020-0255(78)90019-1","article-title":"Fuzzy random variables-I. definitions and theorems","volume":"15","author":"Kwakernaak","year":"1978","journal-title":"Inf. Sci."},{"issue":"1","key":"10.1016\/j.fss.2026.109915_bib0011","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1016\/S0165-0114(98)00035-9","article-title":"Mean-squares Riemann-Stieltjes integrals of fuzzy stochastic processes and their applications","volume":"110","author":"Feng","year":"2000","journal-title":"Fuzzy Sets Syst."},{"issue":"1","key":"10.1016\/j.fss.2026.109915_bib0012","first-page":"61","article-title":"Viability theorems for stochastic inclusions","volume":"15","author":"Kisielewicz","year":"1995","journal-title":"Discus. Math. Differ. Inclusions Contr. Optim."},{"issue":"2","key":"10.1016\/j.fss.2026.109915_bib0013","doi-asserted-by":"crossref","first-page":"480","DOI":"10.1006\/jmaa.1999.6461","article-title":"Stochastic integrals of set-valued processes and fuzzy processes","volume":"236","author":"Kim","year":"1999","journal-title":"J. Math. Anal. Appl."},{"issue":"2","key":"10.1016\/j.fss.2026.109915_bib0014","doi-asserted-by":"crossref","first-page":"409","DOI":"10.1016\/0022-247X(86)90093-4","article-title":"Fuzzy random variables","volume":"114","author":"Kim","year":"1986","journal-title":"J. Math. Anal. Appl."},{"issue":"9","key":"10.1016\/j.fss.2026.109915_bib0015","doi-asserted-by":"crossref","first-page":"949","DOI":"10.1016\/j.fss.2006.12.004","article-title":"Representation theorems, set-valued and fuzzy set-valued It\u00f4 integral","volume":"158","author":"Li","year":"2007","journal-title":"Fuzzy Sets Syst."},{"key":"10.1016\/j.fss.2026.109915_bib0016","series-title":"Stochastic Differential Equations","author":"\u00d8ksendal","year":"1998"},{"key":"10.1016\/j.fss.2026.109915_bib0017","doi-asserted-by":"crossref","unstructured":"E.J. McShane, Stochastic integrals and stochastic functional equations, SIAM J. Appl. Math. 17(1969) 287\u2013306.","DOI":"10.1137\/0117029"},{"key":"10.1016\/j.fss.2026.109915_bib0018","doi-asserted-by":"crossref","unstructured":"Z.R. Pop-Stojanovic, On Mcshane\u2019s belated stochastic integral, SIAM J. Appl. Math. 22(1972) 87\u201392.","DOI":"10.1137\/0122010"},{"key":"10.1016\/j.fss.2026.109915_bib0019","doi-asserted-by":"crossref","unstructured":"T.L. Toh, T.S. Chew, The Riemann approach to stochastic integration using non-uniform meshes, J. Math. Anal. Appl. 280(2003) 133\u2013147.","DOI":"10.1016\/S0022-247X(03)00059-3"},{"key":"10.1016\/j.fss.2026.109915_bib0020","series-title":"Stochastic Calculus and Stochastic Models","author":"McShane","year":"1974"},{"issue":"2","key":"10.1016\/j.fss.2026.109915_bib0021","doi-asserted-by":"crossref","first-page":"276","DOI":"10.1214\/aop\/1176995088","article-title":"A comparison of stochastic integrals","volume":"7","author":"Protter","year":"1979","journal-title":"Ann. Probab."},{"issue":"2","key":"10.1016\/j.fss.2026.109915_bib0022","doi-asserted-by":"crossref","first-page":"352","DOI":"10.2307\/44152280","article-title":"Stochastic integrals of It\u00f4 and Henstock","volume":"18","author":"Xu","year":"1992","journal-title":"Real Anal. Exch."},{"issue":"2","key":"10.1016\/j.fss.2026.109915_bib0023","doi-asserted-by":"crossref","first-page":"495","DOI":"10.14321\/realanalexch.27.2.0495","article-title":"The non-uniform Riemann approach to It\u00f4\u2019s integral","volume":"27","author":"Chew","year":"2001","journal-title":"Real Anal. Exch."},{"key":"10.1016\/j.fss.2026.109915_bib0024","doi-asserted-by":"crossref","unstructured":"G. Muhammad, M. Akram, Fuzzy fractional generalized Bagley-Torvik equation with fuzzy Caputo gH-differentiability, Eng. Appl. Artif. Intell. 133 (2024) 108265.","DOI":"10.1016\/j.engappai.2024.108265"},{"key":"10.1016\/j.fss.2026.109915_bib0025","doi-asserted-by":"crossref","unstructured":"G. Muhammad, M. Akram, N. Hussain, T. Allahviranloo, Fuzzy Langevin fractional delay differential equations under granular derivative, Inf. Sci. 681(2024) 121250.","DOI":"10.1016\/j.ins.2024.121250"},{"key":"10.1016\/j.fss.2026.109915_bib0026","doi-asserted-by":"crossref","unstructured":"G. Muhammad, M. Akram, Fuzzy fractional epidemiological model for middle east respiratory syndrome coronavirus on complex heterogeneous network using Caputo derivative, Inf. Sci. 659(2024) 120046.","DOI":"10.1016\/j.ins.2023.120046"},{"issue":"3","key":"10.1016\/j.fss.2026.109915_bib0027","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1016\/S0888-613X(99)00005-5","article-title":"Solutions to fuzzy integral equations with arbitrary kernels","volume":"20","author":"Friedman","year":"1999","journal-title":"Internation J. Approximate Reason."},{"key":"10.1016\/j.fss.2026.109915_bib0028","doi-asserted-by":"crossref","unstructured":"H.C. Wu, The improper fuzzy Riemann integral and its numerical integration, Inf. Sci. 111(1998) 109\u2013137.","DOI":"10.1016\/S0020-0255(98)00016-4"},{"issue":"3","key":"10.1016\/j.fss.2026.109915_bib0029","doi-asserted-by":"crossref","first-page":"523","DOI":"10.1016\/S0165-0114(99)00057-3","article-title":"On Henstock integral of fuzzy number-valued function (I)","volume":"120","author":"Wu","year":"2001","journal-title":"Fuzzy Sets Syst."},{"issue":"3","key":"10.1016\/j.fss.2026.109915_bib0030","doi-asserted-by":"crossref","first-page":"359","DOI":"10.1016\/S0165-0114(03)00182-9","article-title":"Quadrature rules for integrals of fuzzy-number-valued functions","volume":"145","author":"Bede","year":"2004","journal-title":"Fuzzy Sets Syst."},{"issue":"1","key":"10.1016\/j.fss.2026.109915_bib0031","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1016\/0165-0114(86)90026-6","article-title":"Elementary fuzzy calculus","volume":"18","author":"Goetschel","year":"1986","journal-title":"Fuzzy Sets Syst."},{"key":"10.1016\/j.fss.2026.109915_bib0032","doi-asserted-by":"crossref","unstructured":"Z.T. Gong, L. Wang, The numerical calculus of expectation of fuzzy random variables, Fuzzy Sets Syst. 158(2007) 722\u2013738.","DOI":"10.1016\/j.fss.2006.09.018"},{"key":"10.1016\/j.fss.2026.109915_bib0033","series-title":"The General Theory of Integration","author":"Henstock","year":"1991"},{"key":"10.1016\/j.fss.2026.109915_bib0034","doi-asserted-by":"crossref","first-page":"409","DOI":"10.1016\/0022-247X(86)90093-4","article-title":"Fuzzy random variables","volume":"114","author":"Puri","year":"1986","journal-title":"J. Math. Anal. Appl."},{"issue":"1","key":"10.1016\/j.fss.2026.109915_bib0035","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0165-0114(97)00353-9","article-title":"The fuzzy Riemann integral and its numerical integration","volume":"110","author":"Wu","year":"2000","journal-title":"Fuzzy Sets Syst."},{"issue":"08","key":"10.1016\/j.fss.2026.109915_bib0036","first-page":"1","article-title":"Ito^-Henstock integral of the fuzzy stochastic process","volume":"54","author":"Gong","year":"2019","journal-title":"J. Shandong Univ. (Nat. Sci.)"},{"key":"10.1016\/j.fss.2026.109915_bib0037","series-title":"Stochastic Analytical Foundation","author":"Huang","year":"2001"},{"key":"10.1016\/j.fss.2026.109915_bib0038","series-title":"Applications of Fuzzy Sets to Systems Analysis","author":"Negoita","year":"1975"},{"key":"10.1016\/j.fss.2026.109915_bib0039","doi-asserted-by":"crossref","unstructured":"O. Kaleva, Fuzzy diLerential equations, Fuzzy Sets Syst. 24(1987) 301\u2013317.","DOI":"10.1016\/0165-0114(87)90029-7"},{"key":"10.1016\/j.fss.2026.109915_bib0040","doi-asserted-by":"crossref","unstructured":"M. Ma, On embedding problem of fuzzy number space: Part 4, Fuzzy Sets Syst. 58(1993) 185\u2013193.","DOI":"10.1016\/0165-0114(93)90494-3"},{"key":"10.1016\/j.fss.2026.109915_bib0041","doi-asserted-by":"crossref","unstructured":"C.X. Wu, M. Ma, On embedding problem of fuzzy number space: Part 1, Fuzzy Sets Syst. 44(1991) 33\u201338.","DOI":"10.1016\/0165-0114(91)90030-T"},{"key":"10.1016\/j.fss.2026.109915_bib0042","doi-asserted-by":"crossref","unstructured":"L. Stefanini, B. Bede, Generalized Hukuhara differentiability of interval-valued functions and interval differential equations, Nonlinear Anal. Theory Methods Appl. 71 (2009) 1311\u20131328.","DOI":"10.1016\/j.na.2008.12.005"},{"issue":"2","key":"10.1016\/j.fss.2026.109915_bib0043","first-page":"375","article-title":"Henstock\u2019s version of it\u00f4\u2019s formula","volume":"35","author":"Toh","year":"2009","journal-title":"Real Anal. Exch."},{"key":"10.1016\/j.fss.2026.109915_bib0044","series-title":"Lanzhou Lecture on Henstock Integration","author":"Lee","year":"1989"},{"key":"10.1016\/j.fss.2026.109915_bib0045","doi-asserted-by":"crossref","unstructured":"J. Li, R. Mesiar, E. Pap, E.P. Klement, Convergence theorems for monotone measures, Fuzzy Sets Syst. 281(2015) 103\u2013127.","DOI":"10.1016\/j.fss.2015.05.017"}],"container-title":["Fuzzy Sets and Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0165011426001533?httpAccept=text\/xml","content-type":"text\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0165011426001533?httpAccept=text\/plain","content-type":"text\/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2026,5,27]],"date-time":"2026-05-27T20:01:06Z","timestamp":1779912066000},"score":1,"resource":{"primary":{"URL":"https:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S0165011426001533"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,9]]},"references-count":45,"alternative-id":["S0165011426001533"],"URL":"https:\/\/doi.org\/10.1016\/j.fss.2026.109915","relation":{},"ISSN":["0165-0114"],"issn-type":[{"value":"0165-0114","type":"print"}],"subject":[],"published":{"date-parts":[[2026,9]]},"assertion":[{"value":"Elsevier","name":"publisher","label":"This article is maintained by"},{"value":"The It\u00f4-Henstock integral and its numerical calculation of the fuzzy stochastic process","name":"articletitle","label":"Article Title"},{"value":"Fuzzy Sets and Systems","name":"journaltitle","label":"Journal Title"},{"value":"https:\/\/doi.org\/10.1016\/j.fss.2026.109915","name":"articlelink","label":"CrossRef DOI link to publisher maintained version"},{"value":"article","name":"content_type","label":"Content Type"},{"value":"\u00a9 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.","name":"copyright","label":"Copyright"}],"article-number":"109915"}}