{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,9]],"date-time":"2026-05-09T14:18:03Z","timestamp":1778336283883,"version":"3.51.4"},"reference-count":34,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2025,11,6]],"date-time":"2025-11-06T00:00:00Z","timestamp":1762387200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This paper proposes a fuzzy copula-based optimization framework for modeling dependence structures and financial risk under parameter uncertainty. The parameters of selected copula families are represented as trapezoidal fuzzy numbers, and their \u03b1-cut intervals capture both the support and core ranges of plausible dependence values. This fuzzification transforms the estimation of copula parameters into a fuzzy optimization problem, enhancing robustness against sampling variability. The methodology is empirically applied to gold and oil futures (1 January 2015\u20131 January 2025), comparing symmetric copulas, i.e., Gaussian and Frank and asymmetric copulas, i.e., Clayton, Gumbel and Student-t. The results prove that the fuzzy copula framework provides richer insights than classical point estimation by explicitly expressing uncertainty in dependence measures (Kendall\u2019s \u03c4, Spearman\u2019s \u03c1) and risk indicators (Value-at-Risk, Conditional Value-at-Risk). Rolling-window analyses reveal that fuzzy VaR and fuzzy CVaR effectively capture temporal dependence shifts and tail severity, with fuzzy CVaR consistently producing more conservative risk estimates. This study highlights the potential of fuzzy optimization and fuzzy dependence modeling as powerful tools for quantifying uncertainty and managing extreme co-movements in financial markets.<\/jats:p>","DOI":"10.3390\/sym17111892","type":"journal-article","created":{"date-parts":[[2025,11,6]],"date-time":"2025-11-06T17:51:43Z","timestamp":1762451503000},"page":"1892","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Incorporating Parameter Uncertainty into Copula Models: A Fuzzy Approach"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8536-5636","authenticated-orcid":false,"given":"Irina","family":"Georgescu","sequence":"first","affiliation":[{"name":"Department of Economic Informatics and Cybernetics, Bucharest University of Economic Studies, 0105552 Bucharest, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0154-6617","authenticated-orcid":false,"given":"Jani","family":"Kinnunen","sequence":"additional","affiliation":[{"name":"LUT Business School, LUT University, Yliopistonkatu 31, 53851 Lappeenranta, Finland"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,11,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Dempster, M.A.H. (2002). Correlation and Dependence in Risk Management: Properties and Pitfalls. Risk Management: Value at Risk and Beyond, Cambridge University Press.","DOI":"10.1017\/CBO9780511615337"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1111\/1467-9965.00068","article-title":"Coherent measures of risk","volume":"9","author":"Artzner","year":"1999","journal-title":"Math. Financ."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Just, M., and \u0141uczak, A. (2020). Assessment of Conditional Dependence Structures in Commodity Futures Markets Using Copula-GARCH Models and Fuzzy Clustering Methods. Sustainability, 12.","DOI":"10.3390\/su12062571"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2169","DOI":"10.1080\/02626667.2023.2249459","article-title":"The uncertainty associated with the use of copulas in multivariate analysis","volume":"68","author":"Zhou","year":"2023","journal-title":"Hydrol. Sci. J."},{"key":"ref_5","unstructured":"Buckley, J.J. (2005). Fuzzy Probabilities: New Approach and Applications, Springer."},{"key":"ref_6","unstructured":"Sklar, A. (1959). Fonctions de R\u00e9partition \u00e0 n Dimensions et Leurs Marges, Publications de l\u2019Institut Statistique de l\u2019Universit\u00e9 de Paris."},{"key":"ref_7","first-page":"6047","article-title":"A comprehensive review on the development of copulas in financial field","volume":"45","author":"Ismail","year":"2023","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"453","DOI":"10.1146\/annurev-statistics-040220-101153","article-title":"Vine copula based modeling","volume":"9","author":"Czado","year":"2022","journal-title":"Annu. Rev. Stat. Its Appl."},{"key":"ref_9","unstructured":"Cheng, T., Vatter, T., Nagler, T., and Chen, K. (2025). Vine Copulas as Differentiable Computational Graphs. arXiv."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Dewick, P.R., and Liu, S. (2022). Copula Modelling to Analyse Financial Data. J. Risk Financ. Manag., 15.","DOI":"10.3390\/jrfm15030104"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"965","DOI":"10.1007\/s10614-020-09981-5","article-title":"A new dynamic mixture copula mechanism to examine the nonlinear and asymmetric tail dependence between stock and exchange rate returns","volume":"58","author":"Chang","year":"2021","journal-title":"Comput. Econ."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Saekow, S., Chiawkhun, P., Yamaka, W., Nakharutai, N., and Phetpradap, P. (2025). Global Market Shocks and Tail Risk Spillovers: Evidence from a Copula-Based Contagion Framework. J. Risk Financ. Manag., 18.","DOI":"10.3390\/jrfm18090498"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1080\/07350015.2024.2360592","article-title":"Large Skew-t Copula Models and Asymmetric Dependence in Intraday Equity Returns","volume":"43","author":"Deng","year":"2025","journal-title":"J. Bus. Econ. Stat."},{"key":"ref_14","first-page":"217","article-title":"Construction of Bivariate Asymmetric Copulas","volume":"25","author":"Mukherjee","year":"2018","journal-title":"Commun. Stat. Appl. Methods"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s00180-020-00994-0","article-title":"On tests for symmetry and radial symmetry of bivariate copulas towards testing for ellipticity","volume":"36","author":"Jaser","year":"2021","journal-title":"Comput. Stat."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Billio, M., Frattarolo, L., and Gu\u00e9gan, D. (2022). High-Dimensional Radial Symmetry of Copula Functions: Multiplier Bootstrap vs. Randomization. Symmetry, 14.","DOI":"10.3390\/sym14010097"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1140","DOI":"10.1080\/10618600.2024.2432978","article-title":"Visualization and assessment of copula symmetry","volume":"34","author":"Lee","year":"2025","journal-title":"J. Comput. Graph. Stat."},{"key":"ref_18","unstructured":"Zachariah, S.G., Arshad, M., and Pathak, A.K. (2025). Dependence and uncertainty: Information measures using Tsallis entropy. arXiv."},{"key":"ref_19","first-page":"229","article-title":"A vague multidimensional dependency structure: Conditional versus unconditional fuzzy copula models","volume":"515","author":"Romagnoli","year":"2020","journal-title":"Inf. Sci."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"6058","DOI":"10.1080\/00036846.2021.1935696","article-title":"Vine copulas and fuzzy inference to evaluate the solvency capital requirement of multivariate dependent risks","volume":"53","author":"Araichi","year":"2021","journal-title":"Appl. Econ."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"108466","DOI":"10.1016\/j.fss.2023.01.001","article-title":"Fuzzy Esscher changes of measure and copula invariance in L\u00e9vy markets","volume":"457","author":"Bernardi","year":"2023","journal-title":"Fuzzy Sets Syst."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1221","DOI":"10.1142\/S0219622023500682","article-title":"Hedging salmon price risk based on fuzzy copula-GMM model","volume":"24","author":"Yu","year":"2025","journal-title":"Int. J. Inf. Technol. Decis. Mak."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"5527","DOI":"10.1007\/s40747-023-01002-w","article-title":"A fuzzy rough copula Bayesian network model for solving complex hospital service quality assessment","volume":"9","author":"Li","year":"2023","journal-title":"Complex Intell. Syst."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"338","DOI":"10.1016\/S0019-9958(65)90241-X","article-title":"Fuzzy sets","volume":"8","author":"Zadeh","year":"1965","journal-title":"Inf. Control"},{"key":"ref_25","unstructured":"Dubois, D., and Prade, H. (1980). Fuzzy Sets and Systems: Theory and Applications, Academic Press."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Georgescu, I. (2012). Possibility Theory and the Risk, Springer.","DOI":"10.1007\/978-3-642-24740-8"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"B-141","DOI":"10.1287\/mnsc.17.4.B141","article-title":"Decision-making in a fuzzy environment","volume":"17","author":"Bellman","year":"1970","journal-title":"Manag. Sci."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"2345746","DOI":"10.1155\/2020\/2345746","article-title":"Modelling the Dependency between Inflation and Exchange Rate Using Copula","volume":"2020","author":"Kwofie","year":"2020","journal-title":"J. Probab. Stat."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1093\/biomet\/65.1.141","article-title":"A Model for Association in Bivariate Life Tables and Its Application in Epidemiological Studies of Familial Tendency in Chronic Disease Incidence","volume":"65","author":"Clayton","year":"1978","journal-title":"Biometrika"},{"key":"ref_30","unstructured":"Gumbel, E.J. (1960). Distributions des Valeurs Extr\u00eames en Plusieurs Dimensions, Publications de l\u2019Institut de Statistique de l\u2019Universit\u00e9 de Paris."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"194","DOI":"10.1007\/BF02189866","article-title":"On the simultaneous associativity of F(x,y) and x + y \u2212 F(x,y)","volume":"19","author":"Frank","year":"1979","journal-title":"Aequationes Math."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1111\/j.1751-5823.2005.tb00254.x","article-title":"The t Copula and Related Copulas","volume":"73","author":"Demarta","year":"2005","journal-title":"Int. Stat. Rev."},{"key":"ref_33","first-page":"383","article-title":"Portfolio selection with fuzzy returns","volume":"18","author":"Huang","year":"2007","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Gelman, A., Carlin, J.B., Stern, H.S., Dunson, D.B., Vehtari, A., and Rubin, D.B. (2013). Bayesian Data Analysis, CRC Press. [3rd ed.].","DOI":"10.1201\/b16018"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/11\/1892\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,11,6]],"date-time":"2025-11-06T18:03:26Z","timestamp":1762452206000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/11\/1892"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,11,6]]},"references-count":34,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2025,11]]}},"alternative-id":["sym17111892"],"URL":"https:\/\/doi.org\/10.3390\/sym17111892","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,11,6]]}}}