{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,9]],"date-time":"2026-04-09T21:53:56Z","timestamp":1775771636325,"version":"3.50.1"},"reference-count":75,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2025,4,8]],"date-time":"2025-04-08T00:00:00Z","timestamp":1744070400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This review article is concerned to provide a global context to several works on the fitting of continuous time nonhomogeneous Markov chains with finite state space and also to point out some selected aspects of two techniques previously introduced\u2014estimation and calibration\u2014relevant for applications and used to fit a continuous time Markov chain model to data by the adequate selection of parameters. The denomination estimation suits the procedure better when statistical techniques\u2014e.g., maximum likelihood estimators\u2014are employed, while calibration covers the case where, for instance, some optimisation technique finds a best approximation parameter to ensure good model fitting. For completeness, we provide a short summary of well-known important notions and results formulated for nonhomogeneous Markov chains that, in general, can be transferred to the homogeneous case. Then, as an illustration for the homogeneous case, we present a selected Billingsley\u2019s result on parameter estimation for irreducible chains with finite state space. In the nonhomogeneous case, we quote two recent results, one of the calibration type and the other with more of a statistical flavour. We provide an ample set of bibliographic references so that the reader wanting to pursue her\/his studies will be able to do so more easily and productively.<\/jats:p>","DOI":"10.3390\/axioms14040283","type":"journal-article","created":{"date-parts":[[2025,4,8]],"date-time":"2025-04-08T11:05:02Z","timestamp":1744110302000},"page":"283","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Statistics for Continuous Time Markov Chains, a Short Review"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4991-7568","authenticated-orcid":false,"given":"Manuel L.","family":"Esqu\u00edvel","sequence":"first","affiliation":[{"name":"Department of Mathematics, NOVA School of Science and Technology and NOVA Math, Universidade Nova de Lisboa, 2829-516 Caparica, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4226-1658","authenticated-orcid":false,"given":"Nadezhda P.","family":"Krasii","sequence":"additional","affiliation":[{"name":"NOVA Math, Universidade Nova de Lisboa, 2829-516 Caparica, Portugal"},{"name":"Department of Higher Mathematics, Don State Technical University, Gagarin Square 1, Rostov-on-Don 344000, Russia"}]}],"member":"1968","published-online":{"date-parts":[[2025,4,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1007\/s10182-012-0189-2","article-title":"Multistate models in health insurance","volume":"96","author":"Christiansen","year":"2012","journal-title":"AStA Adv. Stat. Anal."},{"key":"ref_2","first-page":"113","article-title":"On Estimating Transition Intensities of a Markov Process with Aggregate Data of a Certain Type: \u201cOccurrences but No Exposures\u201d","volume":"13","author":"Gill","year":"1986","journal-title":"Scand. J. Stat."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1111\/1467-9965.00114","article-title":"Finding Generators for Markov Chains via Empirical Transition Matrices, with Applications to Credit Ratings","volume":"11","author":"Israel","year":"2001","journal-title":"Math. Financ."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"423","DOI":"10.1016\/S0378-4266(01)00228-X","article-title":"Analyzing rating transitions and rating drift with continuous observations","volume":"26","author":"Lando","year":"2002","journal-title":"J. Bank. Financ."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2603","DOI":"10.1016\/j.jbankfin.2004.06.004","article-title":"Measurement, estimation and comparison of credit migration matrices","volume":"28","author":"Jafry","year":"2004","journal-title":"J. Bank. Financ."},{"key":"ref_6","unstructured":"Inamura, Y. (2006). Estimating Continuous Time Transition Matrices from Discretely Observed Data, Bank of Japan."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1659","DOI":"10.1007\/s00180-020-00978-0","article-title":"Statistical inference for Markov chains with applications to credit risk","volume":"35","author":"Pfeuffer","year":"2020","journal-title":"Comput. Statist."},{"key":"ref_8","unstructured":"Bartholomew, D.J. (1982). Stochastic Models for Social Processes, John Wiley & Sons, Ltd.. [3rd ed.]."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Vassiliou, P.C., and Georgiou, A.C. (2021). Markov and Semi-Markov Chains, Processes, Systems, and Emerging Related Fields. Mathematics, 9.","DOI":"10.3390\/math9192490"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Iosifescu, M., and T\u0103utu, P. (1973). Stochastic Processes and Applications in Biology and Medicine. II: Models, Springer. Biomathematics, Editura Academiei, Bucharest.","DOI":"10.1007\/978-3-642-80753-4"},{"key":"ref_11","first-page":"327","article-title":"Non-homogeneous Markov models for sequential pattern mining of healthcare data","volume":"20","author":"Garg","year":"2009","journal-title":"IMA J. Manag. Math."},{"key":"ref_12","first-page":"92","article-title":"A comparison of time-homogeneous Markov chain and Markov process multi-state models","volume":"2","author":"Wan","year":"2016","journal-title":"Commun. Stat. Case Stud. Data Anal. Appl."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1965","DOI":"10.1002\/sim.9707","article-title":"Non-homogeneous continuous-time Markov chain with covariates: Applications to ambulatory hypertension monitoring","volume":"42","author":"Chang","year":"2023","journal-title":"Stat. Med."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Norris, J.R. (1998). Markov Chains, Cambridge University Press.","DOI":"10.1017\/CBO9780511810633"},{"key":"ref_15","unstructured":"Resnick, S. (1992). Adventures in Stochastic Processes, Birkh\u00e4user Boston, Inc."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Suhov, Y., and Kelbert, M. (2008). Markov chains: A primer in random processes and their applications. Probability and Statistics by Example. II, Cambridge University Press.","DOI":"10.1017\/CBO9780511813641"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Liggett, T.M. (2010). Continuous Time Markov Processes: An Introduction, American Mathematical Society. Graduate Studies in Mathematics.","DOI":"10.1090\/gsm\/113"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Levin, D.A., and Peres, Y. (2017). Markov Chains and Mixing Times, American Mathematical Society. [2nd ed.].","DOI":"10.1090\/mbk\/107"},{"key":"ref_19","unstructured":"Feller, W. (1968). An Introduction to Probability Theory and Its Applications. Vol. I, John Wiley & Sons, Inc.. [3rd ed.]."},{"key":"ref_20","unstructured":"Feller, W. (1971). An Introduction to Probability Theory and Its Applications. Vol. II, John Wiley & Sons, Inc.. [2nd ed.]."},{"key":"ref_21","unstructured":"Brown, D.E., and K\u00f6v\u00e1ry, T. (2006). Theory of Markov Processes, Dover Publications, Inc."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Chung, K.L. (1967). Markov Chains with Stationary Transition Probabilities, Springer, Inc.. [2nd ed.]. Die Grundlehren der Mathematischen Wissenschaften.","DOI":"10.1007\/978-3-642-62015-7"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Freedman, D. (1983). Markov Chains, Springer.","DOI":"10.1007\/978-1-4612-5500-0"},{"key":"ref_24","unstructured":"Stroock, D.W. (2005). An Introduction to Markov Processes, Springer. Graduate Texts in Mathematics."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Kallenberg, O. (2021). Foundations of Modern Probability, Springer. [3rd ed.]. Probability Theory and Stochastic Modelling.","DOI":"10.1007\/978-3-030-61871-1"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Iosifescu, M., and T\u0103utu, P. (1973). Stochastic Processes and Applications in Biology and Medicine. I: Theory, Springer. Biomathematics.","DOI":"10.1007\/978-3-642-80750-3"},{"key":"ref_27","unstructured":"Iosifescu, M. (1980). Finite Markov Processes and Their Applications, Editura Tehnic\u0103. Wiley Series in Probability and Mathematical Statistics."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Rolski, T., Schmidli, H., Schmidt, V., and Teugels, J. (1999). Stochastic Processes for Insurance and Finance, John Wiley & Sons Ltd.","DOI":"10.1002\/9780470317044"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1016\/j.jmaa.2013.09.043","article-title":"On solutions of Kolmogorov\u2019s equations for nonhomogeneous jump Markov processes","volume":"411","author":"Feinberg","year":"2014","journal-title":"J. Math. Anal. Appl."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"582","DOI":"10.1137\/S0040585X97T990630","article-title":"Kolmogorov\u2019s equations for jump Markov processes and their applications to control problems","volume":"66","author":"Feinberg","year":"2022","journal-title":"Theory Probab. Appl."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"587","DOI":"10.1007\/s10479-017-2538-8","article-title":"Kolmogorov\u2019s equations for jump Markov processes with unbounded jump rates","volume":"317","author":"Feinberg","year":"2022","journal-title":"Ann. Oper. Res."},{"key":"ref_32","unstructured":"Slav\u00edk, A. (2007). Product Integration, Its History and Applications, Ne\u010das Center for Mathematical Modeling; Matfyzpress."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Billingsley, P. (1961). Statistical Inference for Markov Processes, University of Chicago Press. Statistical Research Monographs.","DOI":"10.2307\/1401956"},{"key":"ref_34","unstructured":"Doob, J.L. (1953). Stochastic Processes, John Wiley & Sons, Inc."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1007\/BF00531768","article-title":"The imbedding problem for finite Markov chains","volume":"1","author":"Kingman","year":"1962","journal-title":"Z. Wahrscheinlichkeitstheorie Und Verw. Geb."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"345","DOI":"10.1112\/jlms\/s2-8.2.345","article-title":"Some results on the imbedding problem for finite Markov chains","volume":"8","author":"Johansen","year":"1974","journal-title":"J. Lond. Math. Soc."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"032135","DOI":"10.1103\/PhysRevE.93.032135","article-title":"From empirical data to time-inhomogeneous continuous Markov processes","volume":"93","author":"Lencastre","year":"2016","journal-title":"Phys. Rev. E"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"259","DOI":"10.19195\/0208-4147.39.2.2","article-title":"Embedded Markov chain approximations in Skorokhod topologies","volume":"39","year":"2019","journal-title":"Probab. Math. Statist."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"12","DOI":"10.1214\/aoms\/1177705136","article-title":"Statistical methods in Markov chains","volume":"32","author":"Billingsley","year":"1961","journal-title":"Ann. Math. Statist."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"333","DOI":"10.1007\/BF02479763","article-title":"Maximum likelihood estimation for Markov processes","volume":"24","year":"1972","journal-title":"Ann. Inst. Statist. Math."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"74","DOI":"10.1016\/j.spl.2017.09.009","article-title":"Moderate deviation principle for maximum likelihood estimator for Markov processes","volume":"132","year":"2018","journal-title":"Statist. Probab. Lett."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"737","DOI":"10.2307\/1426857","article-title":"Statistical inference for Markov processes when the model is incorrect","volume":"11","author":"Foutz","year":"1979","journal-title":"Adv. Appl. Probab."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1111\/j.1467-9868.2005.00508.x","article-title":"Statistical inference for discretely observed Markov jump processes","volume":"67","author":"Bladt","year":"2005","journal-title":"J. R. Stat. Soc. Ser. B Stat. Methodol."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"970","DOI":"10.1239\/jap\/1067436094","article-title":"Stability and exponential convergence of continuous-time Markov chains","volume":"40","author":"Mitrophanov","year":"2003","journal-title":"J. Appl. Probab."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"1219","DOI":"10.1239\/jap\/1101840568","article-title":"The spectral gap and perturbation bounds for reversible continuous-time Markov chains","volume":"41","author":"Mitrophanov","year":"2004","journal-title":"J. Appl. Probab."},{"key":"ref_46","first-page":"371","article-title":"Stability estimates for continuous-time finite homogeneous Markov chains","volume":"50","author":"Mitrofanov","year":"2005","journal-title":"Teor. Veroyatn. Primen."},{"key":"ref_47","first-page":"159","article-title":"Ergodicity coefficient and perturbation bounds for continuous-time Markov chains","volume":"8","author":"Mitrophanov","year":"2005","journal-title":"Math. Inequal. Appl."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"363","DOI":"10.1017\/S0020268100041731","article-title":"An approach to the study of multiple state models","volume":"111","author":"Waters","year":"1984","journal-title":"J. Inst. Actuar."},{"key":"ref_49","unstructured":"Haberman, S., and Pitacco, E. (1999). Actuarial Models for Disability Insurance, Chapman & Hall\/CRC."},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Olivieri, A., and Pitacco, E. (2015). Introduction to Insurance Mathematics, Springer. [2nd ed.]. European Actuarial Academy (EAA) Series.","DOI":"10.1007\/978-3-319-21377-4"},{"key":"ref_51","doi-asserted-by":"crossref","unstructured":"Mitrophanov, A.Y. (2024). The Arsenal of Perturbation Bounds for Finite Continuous-Time Markov Chains: A Perspective. Mathematics, 12.","DOI":"10.3390\/math12111608"},{"key":"ref_52","doi-asserted-by":"crossref","unstructured":"Esqu\u00edvel, M.L., Krasii, N.P., and Guerreiro, G.R. (2024). Estimation\u2013Calibration of Continuous-Time Non-Homogeneous Markov Chains with Finite State Space. Mathematics, 12.","DOI":"10.3390\/math12050668"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"5475","DOI":"10.1080\/03610926.2020.1734838","article-title":"A class of strong deviation theorems for the sequence of real valued random variables with respect to continuous-state non-homogeneous Markov chains","volume":"50","author":"Zhao","year":"2021","journal-title":"Comm. Statist. Theory Methods"},{"key":"ref_54","doi-asserted-by":"crossref","unstructured":"Esqu\u00edvel, M.L., Guerreiro, G.R., Oliveira, M.C., and Corte Real, P. (2021). Calibration of Transition Intensities for a Multistate Model: Application to Long-Term Care. Risks, 9.","DOI":"10.3390\/risks9020037"},{"key":"ref_55","doi-asserted-by":"crossref","unstructured":"Esqu\u00edvel, M.L., Krasii, N.P., and Guerreiro, G.R. (2021). Open Markov Type Population Models: From Discrete to Continuous Time. Mathematics, 9.","DOI":"10.3390\/math9131496"},{"key":"ref_56","doi-asserted-by":"crossref","unstructured":"Dickson, D.C.M., Hardy, M.R., and Waters, H.R. (2020). Actuarial Mathematics for Life Contingent Risks, Cambridge University Press. [3rd ed.].","DOI":"10.1017\/9781108784184"},{"key":"ref_57","unstructured":"Wolthuis, H. (1994). Life Insurance Mathematics (The Markovian Model), CAIRE."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1017\/S0020268100042633","article-title":"On graduation by mathematical formula","volume":"115","author":"Forfar","year":"1988","journal-title":"J. Inst. Actuar."},{"key":"ref_59","first-page":"407","article-title":"Generalized Linear Models and Actuarial Science","volume":"45","author":"Haberman","year":"1996","journal-title":"J. R. Stat. Society. Ser. D (The Stat.)"},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0167-6687(95)00011-G","article-title":"On the graduations associated with a multiple state model for permanent health insurance","volume":"17","author":"Renshaw","year":"1995","journal-title":"Insur. Math. Econ."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1080\/10920277.2014.978025","article-title":"Multistate actuarial models of functional disability","volume":"19","author":"Fong","year":"2015","journal-title":"N. Am. Actuar. J."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"244","DOI":"10.1080\/03461238.2022.2092891","article-title":"A multi-state model for sick leave and its impact on partial early retirement incentives: The case of the Netherlands","volume":"2023","year":"2023","journal-title":"Scand. Actuar. J."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"700","DOI":"10.1080\/03461238.2020.1724192","article-title":"A multiple state model for the working-age disabled population using cross-sectional data","volume":"2020","author":"Naka","year":"2020","journal-title":"Scand. Actuar. J."},{"key":"ref_64","doi-asserted-by":"crossref","unstructured":"Br\u00e9maud, P. (2020). Markov Chains\u2014Gibbs Fields, Monte Carlo Simulation and Queues, Springer. [2nd ed.]. Texts in Applied Mathematics.","DOI":"10.1007\/978-3-030-45982-6"},{"key":"ref_65","first-page":"141","article-title":"An empirical transition matrix for non-homogeneous Markov chains based on censored observations","volume":"5","author":"Aalen","year":"1978","journal-title":"Scand. J. Statist."},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"509","DOI":"10.1016\/0022-247X(84)90189-6","article-title":"Product integration and solution of ordinary differential equations","volume":"102","author":"Friedman","year":"1984","journal-title":"J. Math. Anal. Appl."},{"key":"ref_67","first-page":"195","article-title":"The product limit estimator as maximum likelihood estimator","volume":"5","author":"Johansen","year":"1978","journal-title":"Scand. J. Stat."},{"key":"ref_68","first-page":"319","article-title":"Nonparametric Estimation of Transition Intensities and Transition Probabilities: A Case Study of a Two-State Markov Process","volume":"38","author":"Keiding","year":"1989","journal-title":"J. R. Stat. Society. Ser. C (Appl. Stat.)"},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1016\/0304-4149(78)90012-1","article-title":"Estimation for discrete time nonhomogeneous Markov chains","volume":"7","author":"Fleming","year":"1978","journal-title":"Stoch. Process. Appl."},{"key":"ref_70","unstructured":"Cramer, R.D. (2001). Parameter Estimation for Discretely Observed Continuous-Time Markov Chains. [Ph.D. Thesis, Rice University]."},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"1773","DOI":"10.1111\/j.1468-0262.2004.00553.x","article-title":"Estimation of continuous-time Markov processes sampled at random time intervals","volume":"72","author":"Duffie","year":"2004","journal-title":"Econometrica"},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"782","DOI":"10.1016\/j.jcp.2006.01.045","article-title":"Fitting timeseries by continuous-time Markov chains: A quadratic programming approach","volume":"217","author":"Crommelin","year":"2006","journal-title":"J. Comput. Phys."},{"key":"ref_73","doi-asserted-by":"crossref","first-page":"1357","DOI":"10.1007\/s00180-022-01273-w","article-title":"On the estimation of partially observed continuous-time Markov chains","volume":"38","author":"Mena","year":"2023","journal-title":"Comput. Statist."},{"key":"ref_74","doi-asserted-by":"crossref","unstructured":"Vassiliou, P.C.G. (2023). Non-Homogeneous Markov Chains and Systems\u2014Theory and Applications, CRC Press.","DOI":"10.1201\/b23204"},{"key":"ref_75","doi-asserted-by":"crossref","first-page":"18","DOI":"10.1007\/s42519-021-00228-6","article-title":"Laws of large numbers for non-homogeneous Markov systems with arbitrary transition probability matrices","volume":"16","author":"Vassiliou","year":"2022","journal-title":"J. Stat. Theory Pract."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/4\/283\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:12:11Z","timestamp":1760029931000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/4\/283"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,4,8]]},"references-count":75,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2025,4]]}},"alternative-id":["axioms14040283"],"URL":"https:\/\/doi.org\/10.3390\/axioms14040283","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,4,8]]}}}