{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T00:43:31Z","timestamp":1759970611378,"version":"build-2065373602"},"reference-count":38,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2025,1,27]],"date-time":"2025-01-27T00:00:00Z","timestamp":1737936000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematics"],"abstract":"<jats:p>This paper presents a comprehensive investigation into the solution existence of stochastic functional integral equations within real separable Banach spaces, emphasizing the establishment of sufficient conditions. Leveraging advanced mathematical tools including probability measures of noncompactness and Petryshyn\u2019s fixed-point theorem adapted for stochastic processes, a robust analytical framework is developed. Additionally, this paper introduces the Euler\u2013Karhunen\u2013Lo\u00e8ve method, which utilizes the Karhunen\u2013Lo\u00e8ve expansion to represent stochastic processes, particularly suited for handling continuous-time processes with an infinite number of random variables. By conducting thorough analysis and computational simulations, which also involve implementing the Euler\u2013Karhunen\u2013Lo\u00e8ve method, this paper effectively highlights the practical relevance of the proposed methodology. Two specific instances, namely, the Delay Cox\u2013Ingersoll\u2013Ross process and modified Black\u2013Scholes with proportional delay model, are utilized as illustrative examples to underscore the effectiveness of this approach in tackling real-world challenges encountered in the realms of finance and stochastic dynamics.<\/jats:p>","DOI":"10.3390\/math13030427","type":"journal-article","created":{"date-parts":[[2025,1,27]],"date-time":"2025-01-27T11:38:49Z","timestamp":1737977929000},"page":"427","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Analytical and Computational Investigations of Stochastic Functional Integral Equations: Solution Existence and Euler\u2013Karhunen\u2013Lo\u00e8ve Simulation"],"prefix":"10.3390","volume":"13","author":[{"given":"Manochehr","family":"Kazemi","sequence":"first","affiliation":[{"name":"Department of Mathematics, Ashtian Branch, Islamic Azad University, Ashtian P.O. Box 39618-13347, Iran"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6372-1556","authenticated-orcid":false,"given":"AliReza","family":"Yaghoobnia","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Lahijan Branch, Islamic Azad University, Lahijan P.O. Box 44169-39515, Iran"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4957-9028","authenticated-orcid":false,"given":"Behrouz Parsa","family":"Moghaddam","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Lahijan Branch, Islamic Azad University, Lahijan P.O. Box 44169-39515, Iran"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8262-1369","authenticated-orcid":false,"given":"Alexandra","family":"Galhano","sequence":"additional","affiliation":[{"name":"Faculdade de Ci\u00eancias Naturais, Engenharias e Tecnologias, Universidade Lus\u00f3fona do Porto, Rua Augusto Rosa 24, 4000-098 Porto, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2025,1,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"7368","DOI":"10.1002\/mma.8975","article-title":"Coupled stochastic systems of Skorokhod type: Well-posedness of a mathematical model and its applications","volume":"46","author":"Muntean","year":"2023","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1214\/21-AOAS1583","article-title":"Fitting stochastic epidemic models to gene genealogies using linear noise approximation","volume":"17","author":"Tang","year":"2023","journal-title":"Ann. Appl. 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