{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,1]],"date-time":"2025-11-01T09:34:42Z","timestamp":1761989682695,"version":"build-2065373602"},"reference-count":22,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2022,9,9]],"date-time":"2022-09-09T00:00:00Z","timestamp":1662681600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["71571053"],"award-info":[{"award-number":["71571053"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This article describes a class of jump-uncertain stochastic control systems, and derives an It\u00f4\u2013Liu formula with jump. We characterize an optimal control law, that satisfies the Hamilton\u2013Jacobi\u2013Bellman equation with jump. Then, this paper deduces the optimal portfolio game under uncertain stochastic financial markets with jump. The information of players is symmetrical. The financial market is constituted of a risk-free asset and a risky asset whose price process is subjected to the jump-uncertain stochastic Black\u2013Scholes model. The game is formulated by two utility maximization problems, each investor tries to maximize his relative utility, which is the weighted average of terminal wealth difference between his terminal wealth and that of his competitor. Finally, the explicit expressions of equilibrium investment strategies and value functions for the constant absolute risk-averse and constant relative risk-averse utility function are derived by using the dynamic programming principle.<\/jats:p>","DOI":"10.3390\/sym14091885","type":"journal-article","created":{"date-parts":[[2022,9,9]],"date-time":"2022-09-09T04:54:41Z","timestamp":1662699281000},"page":"1885","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Uncertain Stochastic Optimal Control with Jump and Its Application in a Portfolio Game"],"prefix":"10.3390","volume":"14","author":[{"given":"Chengyu","family":"Wu","sequence":"first","affiliation":[{"name":"School of Management, Guangdong University of Technology, Guangzhou 510520, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Lu","family":"Yang","sequence":"additional","affiliation":[{"name":"School of Management, Guangdong Polytechnic Normal University, Guangzhou 510665, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Chengke","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Economics and Commence, Guangdong University of Technology, Guangzhou 510520, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,9,9]]},"reference":[{"key":"ref_1","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. 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