{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,5]],"date-time":"2026-05-05T13:34:31Z","timestamp":1777988071398,"version":"3.51.4"},"reference-count":34,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2025,1,17]],"date-time":"2025-01-17T00:00:00Z","timestamp":1737072000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"2020 Shanghai Leading Talents Program of the Shanghai Municipal Human Resources and Social Security Bureau","award":["20LJ2006100"],"award-info":[{"award-number":["20LJ2006100"]}]},{"name":"2020 Shanghai Leading Talents Program of the Shanghai Municipal Human Resources and Social Security Bureau","award":["15ZZ068"],"award-info":[{"award-number":["15ZZ068"]}]},{"name":"2020 Shanghai Leading Talents Program of the Shanghai Municipal Human Resources and Social Security Bureau","award":["15XD1503100"],"award-info":[{"award-number":["15XD1503100"]}]},{"name":"Innovation Program of Shanghai Municipal Education Commission","award":["20LJ2006100"],"award-info":[{"award-number":["20LJ2006100"]}]},{"name":"Innovation Program of Shanghai Municipal Education Commission","award":["15ZZ068"],"award-info":[{"award-number":["15ZZ068"]}]},{"name":"Innovation Program of Shanghai Municipal Education Commission","award":["15XD1503100"],"award-info":[{"award-number":["15XD1503100"]}]},{"name":"Program for Outstanding Academic Leaders in Shanghai City","award":["20LJ2006100"],"award-info":[{"award-number":["20LJ2006100"]}]},{"name":"Program for Outstanding Academic Leaders in Shanghai City","award":["15ZZ068"],"award-info":[{"award-number":["15ZZ068"]}]},{"name":"Program for Outstanding Academic Leaders in Shanghai City","award":["15XD1503100"],"award-info":[{"award-number":["15XD1503100"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this work, we are devoted to discussing a system of fractional stochastic differential variational inequalities with L\u00e9vy jumps (SFSDVI with L\u00e9vy jumps), that comprises both parts, that is, a system of stochastic variational inequalities (SSVI) and a system of fractional stochastic differential equations(SFSDE) with L\u00e9vy jumps. Here it is noteworthy that the SFSDVI with L\u00e9vy jumps consists of both sections that possess a mutual symmetry structure. Invoking Picard\u2019s successive iteration process and projection technique, we obtain the existence of only a solution to the SFSDVI with L\u00e9vy jumps via some appropriate restrictions. In addition, the major outcomes are invoked to deduce that there is only a solution to the spatial-price equilibria system in stochastic circumstances. The main contributions of the article are listed as follows: (a) putting forward the SFSDVI with L\u00e9vy jumps that could be applied for handling different real matters arising from varied domains; (b) deriving the unique existence of solutions to the SFSDVI with L\u00e9vy jumps under a few mild assumptions; (c) providing an applicable instance for spatial-price equilibria system in stochastic circumstances affected with L\u00e9vy jumps and memory.<\/jats:p>","DOI":"10.3390\/sym17010138","type":"journal-article","created":{"date-parts":[[2025,1,17]],"date-time":"2025-01-17T11:24:56Z","timestamp":1737113096000},"page":"138","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["On Symmetrically Stochastic System of Fractional Differential Equations and Variational Inequalities"],"prefix":"10.3390","volume":"17","author":[{"given":"Yue","family":"Zhang","sequence":"first","affiliation":[{"name":"Department of Mathematics, Shanghai Normal University, Shanghai 200234, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3730-1475","authenticated-orcid":false,"given":"Lu-Chuan","family":"Ceng","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Shanghai Normal University, Shanghai 200234, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jen-Chih","family":"Yao","sequence":"additional","affiliation":[{"name":"Research Center for Interneural Computing, China Medical University Hospital, China Medical University, Taichung 404327, Taiwan"},{"name":"Academy of Romanian Scientists, 50044 Bucharest, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yue","family":"Zeng","sequence":"additional","affiliation":[{"name":"Department of Computer Science, University of Illinois at Urbana-Champaign, Champaign, IL 61820, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yun-Yi","family":"Huang","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Shanghai Normal University, Shanghai 200234, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Si-Ying","family":"Li","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Shanghai Normal University, Shanghai 200234, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,1,17]]},"reference":[{"key":"ref_1","first-page":"75","article-title":"A class of stochastic differential variational inequalities with some applications","volume":"24","author":"Zhang","year":"2023","journal-title":"J. Nonlinear Convex. Anal."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"107577","DOI":"10.1016\/j.cnsns.2023.107577","article-title":"Euler scheme for solving a class of stochastic differential variational inequalities with some applications","volume":"127","author":"Zhang","year":"2023","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"345","DOI":"10.1007\/s10107-006-0052-x","article-title":"Differential variational inequalities","volume":"113","author":"Pang","year":"2008","journal-title":"Math. Program."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1007\/s10107-007-0163-z","article-title":"Robust solution of monotone stochastic linear complementarity problems","volume":"117","author":"Chen","year":"2009","journal-title":"Math. Program."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"283","DOI":"10.1023\/B:JOGO.0000026449.29735.3c","article-title":"Time-dependent spatial price equilibrium problem: Existence and stability results for the quantity formulation model","volume":"28","author":"Daniele","year":"2004","journal-title":"J. Global. Optim."},{"key":"ref_6","first-page":"149","article-title":"On generalized global fractional-order composite dynamical systems with set-valued perturbations","volume":"6","author":"Ceng","year":"2022","journal-title":"J. Nonlinear Var. Anal."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2440009","DOI":"10.1142\/S0218348X24400097","article-title":"Solvability and optimal control of a system of semilinear nonlocal fractional evolution inclusions with partial Clarke subdifferential","volume":"32","author":"Ceng","year":"2024","journal-title":"Fractals"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"461","DOI":"10.1007\/s10473-019-0211-9","article-title":"Mixed variational inequalities driven by fractional evolutionary equations","volume":"39","author":"Zeng","year":"2019","journal-title":"Acta Math. Sci."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"118","DOI":"10.1016\/j.camwa.2021.07.003","article-title":"Rothe method and numerical analysis for a new class of fractional differential hemivariational inequality with an application","volume":"98","author":"Weng","year":"2021","journal-title":"Comput. Math. Appl."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"36","DOI":"10.1007\/s00033-018-0929-6","article-title":"A class of fractional differential hemivariational inequalities with application to contact problem","volume":"69","author":"Zeng","year":"2018","journal-title":"Z. Angew. Math. Phys."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"108180","DOI":"10.1016\/j.cnsns.2024.108180","article-title":"On fuzzy fractional differential inclusion driven by variational-hemivariational inequality in Banach spaces","volume":"138","author":"Liang","year":"2024","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"104060","DOI":"10.1016\/j.nonrwa.2023.104060","article-title":"On second order differential inclusion driven by quasi-variational-hemivariational inequalities","volume":"77","author":"Liang","year":"2024","journal-title":"Nonlinear Anal. Real World Appl."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"108373","DOI":"10.1016\/j.cnsns.2024.108373","article-title":"On stochastic fractional differential variational inequalities general system with L\u00e9vy jumps","volume":"140","author":"Ceng","year":"2025","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_14","first-page":"421","article-title":"On approximate controllability for systems of fractional evolution hemivariational inequalities with Riemann-Liouville fractional derivatives","volume":"6","author":"Ceng","year":"2022","journal-title":"J. Nonlinear Var. Anal."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Ceng, L.-C., Fu, Y.-X., Yin, J., He, L., He, L., and Hu, H.-Y. (2021). The solvability of generalized systems of time-dependent hemivariational inequalities enjoying symmetric structure in reflexive Banach spaces. Symmetry, 13.","DOI":"10.3390\/sym13101801"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"5205","DOI":"10.2298\/FIL1815205C","article-title":"Optimal control of feedback control systems governed by systems of evolution hemivariational inequalities","volume":"32","author":"Ceng","year":"2018","journal-title":"Filomat"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Ceng, L.C., Wen, C.F., Liou, Y.C., and Yao, J.C. (2021). A general class of differential hemivariational inequalities systems in reflexive Banach spaces. Mathematics, 9.","DOI":"10.3390\/math9243173"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"022701","DOI":"10.1063\/1.5063514","article-title":"Stochastic fractional differential equations driven by L\u00e9vy noise under Carath\u00e9odory conditions","volume":"60","author":"Abouagwa","year":"2019","journal-title":"J. Math. Phys."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"531","DOI":"10.1515\/fca-2015-0033","article-title":"Decay solutions for a class of fractional differential variational inequalities","volume":"18","author":"Ke","year":"2015","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1016\/j.chaos.2011.12.009","article-title":"Stochastic fractional differential equations: Modeling, method and analysis","volume":"45","author":"Pedjeu","year":"2012","journal-title":"Chaos Solitons Fractals"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1033","DOI":"10.1007\/s11424-015-3075-7","article-title":"Approximate controllability of second-order neutral stochastic differential equations with infinite delay and Poisson jumps","volume":"28","author":"Palanisamy","year":"2015","journal-title":"J. Syst. Sci. Complex."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"110392","DOI":"10.1016\/j.chaos.2020.110392","article-title":"Strong convergence of euler-maruyama scheme to a variable-order fractional stochastic differential equation driven by a multiplicative white noise","volume":"142","author":"Yang","year":"2021","journal-title":"Chaos Solitons Fractals"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"114372","DOI":"10.1016\/j.chaos.2023.114372","article-title":"A stochastic fractional differential variational inequality with L\u00e9vy jump and its application","volume":"178","author":"Zeng","year":"2024","journal-title":"Chaos Solitons Fractals"},{"key":"ref_24","first-page":"559","article-title":"A new fractional nonlinear system driven by a quasi-hemivariational inequality with an application","volume":"22","author":"Weng","year":"2021","journal-title":"J. Nonlinear Convex. Anal."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"318","DOI":"10.1109\/TNN.2004.824252","article-title":"A general projection neural network for solving monotone variational inequalities and related optimization problems","volume":"15","author":"Xia","year":"2004","journal-title":"IEEE Trans. Neural Netw."},{"key":"ref_26","first-page":"199","article-title":"A new class of fractional impulsive differential hemivariational inequalities with an application","volume":"27","author":"Weng","year":"2022","journal-title":"Nonlinear Anal. Model. Control."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Biagini, F., Hu, Y.Z., \u00d8ksendal, B., and Zhang, T.S. (2008). Stochastic Calculus for Fractional Brownian Motion and Applications, Springer.","DOI":"10.1007\/978-1-84628-797-8"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"907","DOI":"10.1016\/j.chaos.2004.03.020","article-title":"Fractional Brownian motions via random walk in the complex plane and via fractional derivative, comparison and further results on their Fokker-Planck equations","volume":"22","author":"Jumarie","year":"2004","journal-title":"Chaos Solitons Fractals"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Yong, J.M., and Zhou, X.Y. (1999). Hamiltonian systems and HJB equations. Stochastic Controls, Springer.","DOI":"10.1007\/978-1-4612-1466-3"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"\u00d8ksendal, B. (2003). An introduction with applications. Stochastic Differential Equations, Springer.","DOI":"10.1007\/978-3-642-14394-6"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"407","DOI":"10.1016\/S0252-9602(15)60012-1","article-title":"Differential inverse variational inequalities in finite dimensional spaces","volume":"35","author":"Li","year":"2015","journal-title":"Acta Math. Sci."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"2159","DOI":"10.1007\/s40995-017-0244-7","article-title":"A new version of Black\u2013Scholes equation presented by time-fractional derivative","volume":"42","author":"Farhadi","year":"2018","journal-title":"Iran. J. Sci. Technol. Trans. A Sci."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"112023","DOI":"10.1016\/j.chaos.2022.112023","article-title":"A Monte-Carlo approach for pricing arithmetic Asian rainbow options under the mixed fractional Brownian motion","volume":"158","author":"Ahmadian","year":"2022","journal-title":"Chaos Solitons Fractals"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"240","DOI":"10.1016\/j.chaos.2016.04.008","article-title":"A very efficient approach for pricing barrier options on an underlying described by the mixed fractional Brownian motion","volume":"87","author":"Ballestra","year":"2016","journal-title":"Chaos Solitons Fractals"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/1\/138\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,8]],"date-time":"2025-10-08T10:31:08Z","timestamp":1759919468000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/1\/138"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,1,17]]},"references-count":34,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2025,1]]}},"alternative-id":["sym17010138"],"URL":"https:\/\/doi.org\/10.3390\/sym17010138","relation":{"has-review":[{"id-type":"doi","id":"10.32388\/R1FHE0","asserted-by":"object"}]},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,1,17]]}}}