{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,18]],"date-time":"2026-01-18T14:17:32Z","timestamp":1768745852373,"version":"3.49.0"},"reference-count":35,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2026,1,17]],"date-time":"2026-01-17T00:00:00Z","timestamp":1768608000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2026,1,17]],"date-time":"2026-01-17T00:00:00Z","timestamp":1768608000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"name":"the National Key Research and Development Program of China","award":["2022YFA1004600"],"award-info":[{"award-number":["2022YFA1004600"]}]},{"DOI":"10.13039\/501100001809","name":"the National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["72342006, 72371253, 72471252, 11931018"],"award-info":[{"award-number":["72342006, 72371253, 72471252, 11931018"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100021171","name":"Guangdong Basic and Applied Basic Research Foundation","doi-asserted-by":"crossref","award":["2023A1515012644"],"award-info":[{"award-number":["2023A1515012644"]}],"id":[{"id":"10.13039\/501100021171","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Discrete Event Dyn Syst"],"published-print":{"date-parts":[[2026,12]]},"DOI":"10.1007\/s10626-026-00424-1","type":"journal-article","created":{"date-parts":[[2026,1,17]],"date-time":"2026-01-17T11:11:23Z","timestamp":1768648283000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Mean-variance optimization of continuous time Markov decision processes"],"prefix":"10.1007","volume":"36","author":[{"given":"Ziyang","family":"Zhou","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Junyu","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2026,1,17]]},"reference":[{"key":"424_CR1","doi-asserted-by":"publisher","unstructured":"Anderson WJ (1991) Continuous-time Markov chains. Springer Berlin. https:\/\/doi.org\/10.1007\/978-1-4612-3038-0","DOI":"10.1007\/978-1-4612-3038-0"},{"key":"424_CR2","volume-title":"Stochastic learning and optimization -a sensitivity-based approach","year":"2007","unstructured":"Cao XR (ed) (2007) Stochastic learning and optimization -a sensitivity-based approach. Springer, New York"},{"key":"424_CR3","doi-asserted-by":"publisher","unstructured":"Chung KL (ed) (1967) Markov Chains with Stationary Transition Probabilities, 2nd ed. Springer, Berlin. https:\/\/doi.org\/10.1007\/978-3-642-62015-7","DOI":"10.1007\/978-3-642-62015-7"},{"key":"424_CR4","doi-asserted-by":"publisher","first-page":"856","DOI":"10.1287\/mnsc.39.7.856","volume":"39","author":"RR Grauer","year":"1993","unstructured":"Grauer RR, Hakansson NH (1993) On the use of mean-variance and quadratic approximations in implementing dynamic investment strategies: a comparison of returns and investment policies. Management Sci 39:856\u2013871. https:\/\/doi.org\/10.1287\/mnsc.39.7.856","journal-title":"Management Sci"},{"issue":"6","key":"424_CR5","doi-asserted-by":"publisher","first-page":"1139","DOI":"10.1109\/TAC.2007.899040","volume":"52","author":"X Guo","year":"2007","unstructured":"Guo X (2007) Constrained optimality for average cost continuous-time Markov decision processes. IEEE Trans Automat Control 52(6):1139\u20131143. https:\/\/doi.org\/10.1109\/TAC.2007.899040","journal-title":"IEEE Trans Automat Control"},{"key":"424_CR6","doi-asserted-by":"publisher","unstructured":"Guo X, Hernn\u00e1ndez-Lerma O (2009) Continuous-time Markov decision processes: theory and applications. Springer New York. https:\/\/doi.org\/10.1007\/978-3-642-02547-1","DOI":"10.1007\/978-3-642-02547-1"},{"issue":"1","key":"424_CR7","doi-asserted-by":"publisher","first-page":"105","DOI":"10.1287\/moor.1100.0477","volume":"36","author":"X Guo","year":"2011","unstructured":"Guo X, Piunovskiy A (2011) Discounted continuous-time markov decision processes with constraints: unbounded transition and loss rates. Math Oper Res 36(1):105\u2013132. https:\/\/doi.org\/10.1287\/moor.1100.0477","journal-title":"Math Oper Res"},{"issue":"9","key":"424_CR8","doi-asserted-by":"publisher","first-page":"2151","DOI":"10.1109\/TAC.2009.2023833","volume":"54","author":"X Guo","year":"2009","unstructured":"Guo X, Song X (2009) Mean-variance criteria for finite continuous-time Markov decision processes. IEEE Trans Automatic Control 54(9):2151\u20132157. https:\/\/doi.org\/10.1109\/TAC.2009.2023833","journal-title":"IEEE Trans Automatic Control"},{"issue":"4","key":"424_CR9","doi-asserted-by":"publisher","first-page":"445","DOI":"10.1007\/s10626-019-00292-y","volume":"29","author":"X Guo","year":"2019","unstructured":"Guo X, Zhang J (2019) Risk-sensitive continuous-time Markov decision processes with unbounded rates and Borel spaces. Discret Event Dyn Syst 29(4):445\u2013471. https:\/\/doi.org\/10.1007\/s10626-019-00292-y","journal-title":"Discret Event Dyn Syst"},{"issue":"2","key":"424_CR10","doi-asserted-by":"publisher","first-page":"177","DOI":"10.1007\/BF02837562","volume":"14","author":"X Guo","year":"2006","unstructured":"Guo X, Hernn\u00e1ndez-Lerma O, Prieto-Rumeau T et al (2006) A survey of recent results on continuous-time Markov decision processes. TOP 14(2):177\u2013246. https:\/\/doi.org\/10.1007\/BF02837562","journal-title":"TOP"},{"issue":"5","key":"424_CR11","doi-asserted-by":"publisher","first-page":"317","DOI":"10.1016\/j.orl.2009.04.005","volume":"37","author":"X Guo","year":"2009","unstructured":"Guo X, Song X, Zhang J (2009) Bias optimality for multichain continuous-time Markov decision processes. Oper Res Lett 37(5):317\u2013321. https:\/\/doi.org\/10.1016\/j.orl.2009.04.005","journal-title":"Oper Res Lett"},{"key":"424_CR12","doi-asserted-by":"publisher","first-page":"539","DOI":"10.1007\/s10626-018-0273-1","volume":"28","author":"Y Huang","year":"2018","unstructured":"Huang Y (2018) Finite horizon continuous-time Markov decision processes with mean and variance criteria. Discret Event Dyn Syst 28:539\u2013564. https:\/\/doi.org\/10.1007\/s10626-018-0273-1","journal-title":"Discret Event Dyn Syst"},{"issue":"4","key":"424_CR13","doi-asserted-by":"publisher","first-page":"675","DOI":"10.1007\/s10626-017-0257-6","volume":"27","author":"H Huo","year":"2017","unstructured":"Huo H, Zou X, Guo X (2017) The risk probability criterion for discounted continuous-time Markov decision processes. Discret Event Dyn Syst 27(4):675\u2013699. https:\/\/doi.org\/10.1007\/s10626-017-0257-6","journal-title":"Discret Event Dyn Syst"},{"key":"424_CR14","doi-asserted-by":"publisher","first-page":"210","DOI":"10.1137\/0310016","volume":"10","author":"P Kakumanu","year":"1972","unstructured":"Kakumanu P (1972) Nondiscounted continuous-time Markov decision processes with countable state and action spaces. SIAM J Control Optim 10:210\u2013220. https:\/\/doi.org\/10.1137\/0310016","journal-title":"SIAM J Control Optim"},{"issue":"6","key":"424_CR15","doi-asserted-by":"publisher","first-page":"1052","DOI":"10.1111\/poms.12853","volume":"27","author":"P Kouvelis","year":"2018","unstructured":"Kouvelis P, Pang Z, Ding Q (2018) Integrated commodity inventory management and financial hedging: a dynamic mean variance analysis. Prod Oper Manag 27(6):1052\u20131073. https:\/\/doi.org\/10.1111\/poms.12853","journal-title":"Prod Oper Manag"},{"key":"424_CR16","doi-asserted-by":"publisher","first-page":"96","DOI":"10.1109\/9.898698","volume":"46","author":"ME Lewis","year":"2001","unstructured":"Lewis ME, Puterman ML (2001) A probabilistic analysis of bias optimality in unichain Markov decision processes. IEEE Trans Automatic Control 46:96\u2013100. https:\/\/doi.org\/10.1109\/9.898698","journal-title":"IEEE Trans Automatic Control"},{"key":"424_CR17","doi-asserted-by":"publisher","unstructured":"Lothar B (2003) From Markov jump processes to spatial queues. Springer Dordrecht. https:\/\/doi.org\/10.1007\/978-94-010-0239-4","DOI":"10.1007\/978-94-010-0239-4"},{"key":"424_CR18","doi-asserted-by":"publisher","first-page":"569","DOI":"10.48550\/arXiv.2206.07376","volume":"75","author":"X Ma","year":"2022","unstructured":"Ma X, Ma S, Xia L et al (2022) Mean-semivariance policy optimization via risk-averse reinforcement learning. J Artif Intell Res 75:569\u2013595. https:\/\/doi.org\/10.48550\/arXiv.2206.07376","journal-title":"J Artif Intell Res"},{"issue":"1","key":"424_CR19","doi-asserted-by":"publisher","first-page":"77","DOI":"10.2307\/2975974","volume":"7","author":"H Markowitz","year":"1952","unstructured":"Markowitz H (1952) Portfolio selection. J Finance 7(1):77\u201391. https:\/\/doi.org\/10.2307\/2975974","journal-title":"J Finance"},{"issue":"3","key":"424_CR20","doi-asserted-by":"publisher","first-page":"552","DOI":"10.1016\/0022-247X(68)90194-7","volume":"22","author":"BL Miller","year":"1968","unstructured":"Miller BL (1968) Finite state continuous time Markov decision processes with an infinite planning horizon. J Math Anal Appl 22(3):552\u2013569. https:\/\/doi.org\/10.1016\/0022-247X(68)90194-7","journal-title":"J Math Anal Appl"},{"issue":"2","key":"424_CR21","doi-asserted-by":"publisher","first-page":"215","DOI":"10.1086\/295078","volume":"41","author":"J Mossin","year":"1968","unstructured":"Mossin J (1968) Optimal multiperiod portfolio policies. J Business 41(2):215\u2013229","journal-title":"J Business"},{"issue":"5","key":"424_CR22","doi-asserted-by":"publisher","first-page":"2032","DOI":"10.1137\/10081366X","volume":"49","author":"A Piunovskiy","year":"2011","unstructured":"Piunovskiy A, Zhang Y (2011) Discounted continuous-time markov decision processes with unbounded rates: the dynamic programming approach. SIAM J Control Optim 49(5):2032\u20132061. https:\/\/doi.org\/10.1137\/10081366X","journal-title":"SIAM J Control Optim"},{"key":"424_CR23","doi-asserted-by":"publisher","unstructured":"Piunovskiy A, Zhang Y (eds) (2020) Continuous-time Markov decision processes borel space models and general control strategies. Springer Switzerland. https:\/\/doi.org\/10.1007\/978-3-030-54987-9","DOI":"10.1007\/978-3-030-54987-9"},{"key":"424_CR24","doi-asserted-by":"publisher","first-page":"52","DOI":"10.1137\/S036301290343432","volume":"45","author":"T Prieto-Rumeau","year":"2003","unstructured":"Prieto-Rumeau T, Hernn\u00e1ndez-Lerma O (2003) Bias optimality for continuous-time controlled Markov chains. SIAM J Control Optim 45:52\u201373. https:\/\/doi.org\/10.1137\/S036301290343432","journal-title":"SIAM J Control Optim"},{"key":"424_CR25","doi-asserted-by":"publisher","unstructured":"Puterman ML (ed) (1994) Markov decision processes: discrete stochastic dynamic programming. John & Wiley Sons New York. https:\/\/doi.org\/10.1002\/9780470316887","DOI":"10.1002\/9780470316887"},{"key":"424_CR26","doi-asserted-by":"publisher","first-page":"239","DOI":"10.2307\/1926559","volume":"51","author":"PA Samuelson","year":"1969","unstructured":"Samuelson PA (1969) Lifetime portfolio selection by dynamic stochastic programming. Rev Econom Statist 51:239\u2013246. https:\/\/doi.org\/10.2307\/1926559","journal-title":"Rev Econom Statist"},{"key":"424_CR27","doi-asserted-by":"publisher","first-page":"181","DOI":"10.1093\/imaman\/dpab009","volume":"33","author":"R Schlosser","year":"2021","unstructured":"Schlosser R (2021) Heuristic mean-variance optimization in Markov decision processes using state-dependent risk aversion. IMA J Manag Math 33:181\u2013199. https:\/\/doi.org\/10.1093\/imaman\/dpab009","journal-title":"IMA J Manag Math"},{"key":"424_CR28","doi-asserted-by":"publisher","unstructured":"Schulman J, Levine S, Moritz P et\u00a0al (2015) Trust region policy optimization. Comput Sci 1889\u20131897. https:\/\/doi.org\/10.48550\/arXiv.1502.05477","DOI":"10.48550\/arXiv.1502.05477"},{"key":"424_CR29","doi-asserted-by":"publisher","first-page":"269","DOI":"10.1016\/j.automatica.2016.06.018","volume":"73","author":"L Xia","year":"2016","unstructured":"Xia L (2016) Optimization of Markov decision processes under the variance criterion. Automatica 73:269\u2013278. https:\/\/doi.org\/10.1016\/j.automatica.2016.06.018","journal-title":"Automatica"},{"key":"424_CR30","doi-asserted-by":"publisher","first-page":"76","DOI":"10.1016\/j.automatica.2017.11.012","volume":"88","author":"L Xia","year":"2018","unstructured":"Xia L (2018a) Mean-variance optimization of discrete time discounted Markov decision processes. Automatica 88:76\u201382. https:\/\/doi.org\/10.1016\/j.automatica.2017.11.012","journal-title":"Automatica"},{"key":"424_CR31","doi-asserted-by":"publisher","first-page":"63","DOI":"10.1007\/s10626-017-0258-5","volume":"28","author":"L Xia","year":"2018","unstructured":"Xia L (2018b) Variance minimization of parameterized Markov decision processes. Discrete Event Dynamic Systems 28:63\u201381. https:\/\/doi.org\/10.1007\/s10626-017-0258-5","journal-title":"Discrete Event Dynamic Systems"},{"issue":"12","key":"424_CR32","doi-asserted-by":"publisher","first-page":"2808","DOI":"10.1111\/poms.13252","volume":"29","author":"L Xia","year":"2020","unstructured":"Xia L (2020) Risk-sensitive Markov decision processes with combined metrics of mean and variance. Prod Oper Manag 29(12):2808\u20132827. https:\/\/doi.org\/10.1111\/poms.13252","journal-title":"Prod Oper Manag"},{"issue":"3","key":"424_CR33","doi-asserted-by":"publisher","first-page":"349","DOI":"10.1109\/TAC.2004.824479","volume":"49","author":"G Yin","year":"2004","unstructured":"Yin G, Zhou XY (2004) Markowitz\u2019s mean-variance portfolio selection with regime switching: from discrete-time models to their continuous-time limits. IEEE Trans Automatic Control 49(3):349\u2013360. https:\/\/doi.org\/10.1109\/TAC.2004.824479","journal-title":"IEEE Trans Automatic Control"},{"issue":"7","key":"424_CR34","doi-asserted-by":"publisher","first-page":"1628","DOI":"10.1016\/j.automatica.2009.03.009","volume":"44","author":"J Zhang","year":"2009","unstructured":"Zhang J, Cao XR (2009) Continuous-time Markov decision processes with $$n$$th-bias optimality criteria. Automatica 44(7):1628\u20131638. https:\/\/doi.org\/10.1016\/j.automatica.2009.03.009","journal-title":"Automatica"},{"key":"424_CR35","doi-asserted-by":"publisher","first-page":"19","DOI":"10.1007\/s002450010003","volume":"42","author":"XY Zhou","year":"2000","unstructured":"Zhou XY, Li D (2000) Continuous-time mean-variance portfolio selection: a stochastic LQ framework. Appl Math Optim 42:19\u201333. https:\/\/doi.org\/10.1007\/s002450010003","journal-title":"Appl Math Optim"}],"container-title":["Discrete Event Dynamic Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10626-026-00424-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10626-026-00424-1","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10626-026-00424-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,1,17]],"date-time":"2026-01-17T11:11:25Z","timestamp":1768648285000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10626-026-00424-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,1,17]]},"references-count":35,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2026,12]]}},"alternative-id":["424"],"URL":"https:\/\/doi.org\/10.1007\/s10626-026-00424-1","relation":{},"ISSN":["0924-6703","1573-7594"],"issn-type":[{"value":"0924-6703","type":"print"},{"value":"1573-7594","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,1,17]]},"assertion":[{"value":"30 July 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"1 December 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 January 2026","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}],"article-number":"1"}}