{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:04:59Z","timestamp":1760148299183,"version":"build-2065373602"},"reference-count":44,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2023,4,21]],"date-time":"2023-04-21T00:00:00Z","timestamp":1682035200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100011789","name":"Natural Science Foundation of Jilin Province","doi-asserted-by":"publisher","award":["YDZJ202201ZYTS516"],"award-info":[{"award-number":["YDZJ202201ZYTS516"]}],"id":[{"id":"10.13039\/501100011789","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this paper, we propose a new discrete-time risk model of an insurance portfolio with stochastic premiums, in which the temporal dependence among the premium numbers of consecutive periods is fitted by the first-order integer-valued autoregressive (INAR(1)) process and the temporal dependence among the claim numbers of consecutive periods is described by the integer-valued moving average (INMA(1)) process. To measure the risk of the model quantitatively, we study the explicit expression for a function whose solution is defined as the Lundberg adjustment coefficient and give the Lundberg approximation formula for the infinite-time ruin probability. In the case of heavy-tailed claim sizes, we establish the asymptotic formula for the finite-time ruin probability via the large deviations of the aggregate claims. Two numerical examples are provided in order to illustrate our theoretical findings.<\/jats:p>","DOI":"10.3390\/e25040698","type":"journal-article","created":{"date-parts":[[2023,4,21]],"date-time":"2023-04-21T04:48:11Z","timestamp":1682052491000},"page":"698","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Ruin Analysis on a New Risk Model with Stochastic Premiums and Dependence Based on Time Series for Count Random Variables"],"prefix":"10.3390","volume":"25","author":[{"given":"Lihong","family":"Guan","sequence":"first","affiliation":[{"name":"School of Science, Changchun University, Changchun 130022, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiaohong","family":"Wang","sequence":"additional","affiliation":[{"name":"Mathematics and Computer College, Jilin Normal University, Siping 136000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,4,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Asmussen, S., and Albrecher, H. (2010). Ruin Probabilities, World Scientiffic. [2nd ed.].","DOI":"10.1142\/7431"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1017\/S0269964803172026","article-title":"Martingale method for ruin probability in an autoregressive model with constant interest rate","volume":"17","author":"Yang","year":"2003","journal-title":"Probab. Eng. Informational Sci."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1016\/j.insmatheco.2021.10.006","article-title":"Risk aggregation under dependence uncertainty and an order constraint","volume":"102","author":"Chen","year":"2022","journal-title":"Insur. Math. Econ."},{"key":"ref_4","first-page":"219","article-title":"Value-at-Risk, Tail Value-at-Risk and upper tail transform of the sum of two counter-monotonic random variables","volume":"2022","author":"Hanbali","year":"2022","journal-title":"Scand. Actuar. J."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"96","DOI":"10.1016\/j.insmatheco.2022.01.004","article-title":"A bivariate Laguerre expansions approach for joint ruin probabilities in a two-dimensional insurance risk process","volume":"103","author":"Albrecher","year":"2022","journal-title":"Insur. Math. Econ."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Strietzel, P.L., and Heinrich, H.E. (2022). Optimal dividends for a two-dimensional risk model with simultaneous ruin of both branches. Risk, 10.","DOI":"10.3390\/risks10060116"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"5728","DOI":"10.1080\/03610926.2020.1845737","article-title":"Optimal dividends and reinsurance with capital injection under thinning dependence","volume":"51","author":"Chen","year":"2022","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Wang, F.D., and Liang, Z.B. (2022). Optimal per-loss reinsurance for a risk model with a thinning-dependence structure. Mathematics, 10.","DOI":"10.3390\/math10234621"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1016\/0167-6687(82)90011-7","article-title":"Ruin theory in the linear model","volume":"1","author":"Gerber","year":"1982","journal-title":"Insur. Math. Econ."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1016\/j.insmatheco.2003.11.005","article-title":"Heterogeneous INAR(1) model with application to car insurance","volume":"34","author":"Gourieroux","year":"2004","journal-title":"Insur. Math. Econ."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"123","DOI":"10.2143\/AST.40.1.2049221","article-title":"Discrete-time risk models based on time series for count random variables","volume":"40","author":"Cossette","year":"2010","journal-title":"ASTIN Bull."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1016\/j.insmatheco.2010.08.007","article-title":"Risk models based on time series for count random variables","volume":"48","author":"Cossette","year":"2011","journal-title":"Insur. Math. Econ."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"15305","DOI":"10.1080\/03610926.2012.729636","article-title":"An approximation model of the collective risk model with INAR(1) claim process","volume":"43","author":"Shi","year":"2014","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"455","DOI":"10.1080\/03461238.2013.849615","article-title":"Optimal reinsurance under adjustment coefficient measure in a discrete risk model based on Poisson MA(1) process","volume":"2015","author":"Zhang","year":"2015","journal-title":"Scand. Actuar. J."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"412","DOI":"10.1080\/03461238.2017.1371067","article-title":"Risk model based on the first-order integer-valued moving average process with compound Poisson distributed innovations","volume":"2018","author":"Hu","year":"2018","journal-title":"Scand. Actuar. J."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"3985","DOI":"10.1080\/03610926.2019.1594297","article-title":"Risk aggregation with dependence and overdispersion based on the compound Poisson INAR(1) process","volume":"49","author":"Chen","year":"2020","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1080\/03461238.2021.1937305","article-title":"On the analysis of a discrete-time risk model with INAR(1) processes","volume":"2022","author":"Guan","year":"2022","journal-title":"Scand. Actuar. J."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"489","DOI":"10.1137\/S0040585X9797987","article-title":"The Cram\u00e9r-Lundberg model with stochastic premium process","volume":"47","author":"Boikov","year":"2003","journal-title":"Theory Probab. Its Appl."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"467","DOI":"10.1007\/s11464-007-0029-y","article-title":"Ruin problems with stochastic premium stochastic return on investments","volume":"2","author":"Wang","year":"2017","journal-title":"Front. Math. China"},{"key":"ref_20","first-page":"1852","article-title":"The expected discounted penalty function under a risk model with stochastic income","volume":"215","author":"Sendova","year":"2019","journal-title":"Appl. Math. Comput."},{"key":"ref_21","first-page":"6144","article-title":"The expected discounted penalty function under a renewal risk model with stochastic income","volume":"217","author":"Zhao","year":"2012","journal-title":"Appl. Math. Comput."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"5686","DOI":"10.1080\/03610926.2019.1620782","article-title":"Estimating the Gerber-Shiu function under a risk model with stochastic income by Laguerre series expansion","volume":"49","author":"Su","year":"2020","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_23","first-page":"489","article-title":"Simple approximations for the ruin probability in the risk model with stochastic premiums and a constant dividend strategy","volume":"7","author":"Ragulina","year":"2020","journal-title":"Mod. Stochastics Theory Appl."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"969","DOI":"10.1007\/s10479-021-03937-0","article-title":"On a double barrier hybrid dividend strategy in a compound Poisson risk model with stochastic income","volume":"315","author":"Dibu","year":"2022","journal-title":"Ann. Oper. Res."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1016\/j.insmatheco.2021.02.005","article-title":"An insurance risk process with a generalized income process: A solvency analysis","volume":"98","author":"Wang","year":"2021","journal-title":"Insur. Math. Econ."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"20823","DOI":"10.3934\/math.20221141","article-title":"A discrete-time dual risk model with dependence based on a Poisson INAR(1) process","volume":"7","author":"Guan","year":"2022","journal-title":"AIMS Math."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1007\/s10182-008-0072-3","article-title":"Thinning operations for modeling time series of counts\u2014A survey","volume":"92","year":"2008","journal-title":"AStA Adv. Stat. Anal."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"590","DOI":"10.1177\/1471082X15584701","article-title":"Thinning-based models in the analysis of integer-valued time series: A review","volume":"15","author":"Scotto","year":"2015","journal-title":"Stat. Model."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Wei\u00df, C.H. (2018). An Introduction to Discrete-Valued Time Series, John Wiley and Sons Ltd.","DOI":"10.1002\/9781119097013"},{"key":"ref_30","first-page":"e1502","article-title":"Stationary count time series models","volume":"11","year":"2021","journal-title":"WIREs Comput. Stat."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1111\/j.1467-9892.1987.tb00438.x","article-title":"First-order integer-valued autoregressive (INAR(1)) process","volume":"8","author":"Alzaid","year":"1987","journal-title":"J. Time Ser. Anal."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1016\/S0167-6687(01)00063-4","article-title":"Asymptotic ruin probabilities for risk processes with dependent increments","volume":"28","author":"Pflug","year":"2001","journal-title":"Insur. Math. Econ."},{"key":"ref_33","first-page":"293","article-title":"Large deviations of heavy-tailed random sums with applications in insurance and finance","volume":"24","author":"Mikosch","year":"1997","journal-title":"J. Appl. Probab."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.insmatheco.2020.12.003","article-title":"Precise large deviations of aggregate claims with arbitrary dependence between claim sizes and waiting times","volume":"97","author":"Chen","year":"2021","journal-title":"Insur. Math. Econ."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"109365","DOI":"10.1016\/j.spl.2022.109365","article-title":"Precise large deviations in a bidimensional risk model with arbitrary dependence between claim-size vectors and waiting times","volume":"184","author":"Fu","year":"2022","journal-title":"Stat. Probab. Lett."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"3822","DOI":"10.1016\/j.amc.2011.09.030","article-title":"Ruin problems for an autoregressive risk model with dependent rates of interest","volume":"218","author":"Cheng","year":"2011","journal-title":"Appl. Math. Comput."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"669","DOI":"10.1239\/jap\/1409932666","article-title":"Uniform asymptotics for discounted aggregate claims in dependent risk models","volume":"51","author":"Yang","year":"2014","journal-title":"J. Appl. Probab."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"3761","DOI":"10.1080\/03610926.2020.1801740","article-title":"Asymptotic estimates for finite-time ruin probability in a discrete-time risk model with dependence structures and CMC simulations","volume":"51","author":"Jing","year":"2022","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1541","DOI":"10.3934\/jimo.2021032","article-title":"The finite-time ruin probability of a risk model with a general counting process and stochastic return","volume":"18","author":"Xun","year":"2022","journal-title":"J. Ind. Manag. Optim."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"140","DOI":"10.1186\/s13660-016-1080-6","article-title":"Precise large deviations of aggregate claims in a discrete-time risk model with Poisson ARCH claim-number process","volume":"2016","author":"Yu","year":"2016","journal-title":"J. Inequalities Appl."},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Embrechts, P., Kl\u00fcppelberg, C., and Mikosch, T. (1997). Modelling Extremal Events: For Insurance and Finance, Springer Science & Business Media.","DOI":"10.1007\/978-3-642-33483-2"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1239\/jap\/1077134670","article-title":"Precise large deviations for sums of random variables with consistently varying tails","volume":"41","author":"Ng","year":"2004","journal-title":"J. Appl. Probab."},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Dembo, A., and Zeitouni, O. (1998). Large Deviations Techniques and Applications, Springer. [2nd ed.].","DOI":"10.1007\/978-1-4612-5320-4"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1515\/mcma.2002.8.2.111","article-title":"Simulation of ruin probabilities for risk processes of Markovian type","volume":"8","author":"Albrecher","year":"2002","journal-title":"Monte Carlo Methods Appl."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/4\/698\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:20:26Z","timestamp":1760124026000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/4\/698"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,4,21]]},"references-count":44,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2023,4]]}},"alternative-id":["e25040698"],"URL":"https:\/\/doi.org\/10.3390\/e25040698","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2023,4,21]]}}}