{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,31]],"date-time":"2026-01-31T07:10:52Z","timestamp":1769843452505,"version":"3.49.0"},"reference-count":13,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2023,9,13]],"date-time":"2023-09-13T00:00:00Z","timestamp":1694563200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Open Access Publishing Fund of the Free University of Bozen-Bolzano"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We propose and examine a model expressed by stochastic differential equations for the evolution of a complex system. We refer in particular to a market society, in which the state of each individual is identified by the amount of money at his\/her disposal. The evolution of such a system over time is described by suitable equations that link the instantaneous changes in the probability of each state with the probable outcomes of pairwise interactions between elements of the system. In the context at hand, these pairwise interactions simply represent money exchanges, due to the sales and purchases of goods and services. In this paper, unlike the usual method in the literature, the interaction frequencies and the consequent probabilities of passing from one state to another are not considered as assigned once and for all but are supposed to be randomly variable. This choice, as also shown by several numerical simulations, seems likely to have fruitful consequences, especially for a more realistic representation of economic issues and phenomena.<\/jats:p>","DOI":"10.3390\/sym15091751","type":"journal-article","created":{"date-parts":[[2023,9,13]],"date-time":"2023-09-13T05:08:21Z","timestamp":1694581701000},"page":"1751","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Modelling a Market Society with Stochastically Varying Money Exchange Frequencies"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6086-8395","authenticated-orcid":false,"given":"Maria Letizia","family":"Bertotti","sequence":"first","affiliation":[{"name":"Faculty of Engineering, Free University of Bozen-Bolzano, 39100 Bolzano, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Bruno","family":"Carbonaro","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Physics, University of Campania \u201cL. Vanvitelli\u201d, 81100 Caserta, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8822-3038","authenticated-orcid":false,"given":"Marco","family":"Menale","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Physics, University of Campania \u201cL. Vanvitelli\u201d, 81100 Caserta, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,9,13]]},"reference":[{"key":"ref_1","first-page":"752","article-title":"Modelling taxation and redistribution: A discrete active particle kinetic approach","volume":"217","author":"Bertotti","year":"2010","journal-title":"Appl. Math. Comput."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1140\/epjb\/e2007-00343-8","article-title":"Kinetic exchange models for income and wealth distributions","volume":"60","author":"Chatterjee","year":"2007","journal-title":"Eur. Phys. J. B"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"056103","DOI":"10.1103\/PhysRevE.78.056103","article-title":"Kinetic equations modelling wealth redistribution: A comparison of approaches","volume":"78","author":"Matthes","year":"2008","journal-title":"Phys. Rev. E"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"4151","DOI":"10.1016\/j.physa.2009.06.038","article-title":"Microeconomics of the ideal gas like market models","volume":"388","author":"Chakrabarti","year":"2009","journal-title":"Physica A"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1703","DOI":"10.1103\/RevModPhys.81.1703","article-title":"Colloquium: Statistical mechanics of money, wealth, and income","volume":"81","author":"Yakovenko","year":"2009","journal-title":"Rev. Mod. Phys."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"042804","DOI":"10.1103\/PhysRevE.89.042804","article-title":"Kinetics of wealth and the Pareto law","volume":"89","author":"Boghosian","year":"2014","journal-title":"Phys. Rev. E"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"287","DOI":"10.1515\/ijnsns-2017-0228","article-title":"Stochastic models with multiplicative noise for economic inequality and mobility","volume":"22","author":"Bertotti","year":"2021","journal-title":"Int. J. of Nonlinear Sci. Numer. Simul."},{"key":"ref_8","unstructured":"Oksendal, B. (2007). Stochastic Differential Equations: An Introduction with Applications, Springer. [6th ed.]."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Evans, L.C. (2013). An Introduction to Stochastic Differential Equations, AMS.","DOI":"10.1090\/mbk\/082"},{"key":"ref_10","unstructured":"Herzog, F. (2023, July 08). Stochastic Differential Equations. Available online: https:\/\/ethz.ch\/content\/dam\/ethz\/special-interest\/mavt\/dynamic-systems-n-control\/idsc-dam\/Lectures\/Stochastic-Systems\/SDE.pdf."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Higham, D.J., and Kloeden, P.E. (2020). An Introduction to the Numerical Simulation of Stochastic Differential Equations, SIAM.","DOI":"10.1137\/1.9781611976434"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Lovric, M. (2011). International Encyclopedia of Statistical Science, Springer.","DOI":"10.1007\/978-3-642-04898-2"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Bianca, C., Carbonaro, B., and Menale, M. (2020). On the Cauchy problem of vectorial thermostatted kinetic frameworks. Symmetry, 12.","DOI":"10.3390\/sym12040517"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/9\/1751\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T20:50:00Z","timestamp":1760129400000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/9\/1751"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,9,13]]},"references-count":13,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2023,9]]}},"alternative-id":["sym15091751"],"URL":"https:\/\/doi.org\/10.3390\/sym15091751","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,9,13]]}}}