{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:43:30Z","timestamp":1760060610128,"version":"build-2065373602"},"reference-count":44,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2025,9,11]],"date-time":"2025-09-11T00:00:00Z","timestamp":1757548800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"FCT\u2014Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","award":["UIDB\/00297\/2020","UIDP\/00297\/2020"],"award-info":[{"award-number":["UIDB\/00297\/2020","UIDP\/00297\/2020"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Risks"],"abstract":"<jats:p>In the dual risk model, while the ultimate ruin probability has an exact and straightforward formula, the mathematics becomes significantly more complex when considering a finite time horizon, and the literature on this topic is scarce. As a result, there is a need for numerical approximations. To address this, we develop two numerical algorithms that can accommodate a wide range of distributions for the amount of individual earnings with minimal adjustments. These algorithms are grounded in the methodologies proposed by Cardoso and Eg\u00eddio dos Reis (2002) and De Vylder and Goovaerts (1988), which involve approximating the continuous risk process with a discrete-time Markov chain framework. We work out some examples, providing approximate values for the density of the time to ruin, and we compare, in the long run, our approximations with the exact values for the ultimate ruin probability to evaluate their accuracy. We also benchmark our results against the few existing figures available in the literature. Our findings suggest that the proposed approaches offer an efficient and flexible methodology for computing finite-time ruin probabilities in the dual risk model.<\/jats:p>","DOI":"10.3390\/risks13090174","type":"journal-article","created":{"date-parts":[[2025,9,11]],"date-time":"2025-09-11T14:23:50Z","timestamp":1757600630000},"page":"174","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Numerical Calculation of Finite-Time Ruin Probabilities in the Dual Risk Model"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0715-7954","authenticated-orcid":false,"given":"Rui M. R.","family":"Cardoso","sequence":"first","affiliation":[{"name":"Center for Mathematics and Applications (NOVA Math) and Department of Mathematics, NOVA School of Science and Technology (NOVA FCT), Universidade Nova de Lisboa, Campus de Caparica, 2829-516 Caparica, Portugal"}]},{"given":"Andressa C. O.","family":"Melo","sequence":"additional","affiliation":[{"name":"Department of Mathematics, NOVA School of Science and Technology (NOVA FCT), Universidade Nova de Lisboa, Campus de Caparica, 2829-516 Caparica, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2025,9,11]]},"reference":[{"key":"ref_1","first-page":"906","article-title":"Dividend problems in the dual risk model","volume":"53","author":"Afonso","year":"2013","journal-title":"Insurance: Mathematics and Economics"},{"key":"ref_2","first-page":"313","article-title":"Blockchain mining in pools: Analyzing the trade-off between profitability and ruin","volume":"105","author":"Albrecher","year":"2022","journal-title":"Insurance: Mathematics and Economics"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Asmussen, Soren, and Albrecher, Hansj\u00f6rg (2010). Ruin Probabilities, World Scientific Publishing Company. [2nd ed.]. Advanced Series On Statistical Science And Applied Probability.","DOI":"10.1142\/7431"},{"key":"ref_4","first-page":"111","article-title":"Optimal dividends in the dual model","volume":"41","author":"Avanzi","year":"2007","journal-title":"Insurance: Mathematics and Economics"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1007\/s00186-007-0181-x","article-title":"Optimizing venture capital investment in a jump diffusion model","volume":"67","author":"Bayraktar","year":"2008","journal-title":"Mathematical Methods of Operations Research"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1162","DOI":"10.1080\/03610926.2022.2093910","article-title":"On the evaluation of ruin probabilities in a generalized dual binomial risk model using markov property","volume":"53","author":"Bazyari","year":"2024","journal-title":"Communications in Statistics\u2014Theory and Methods"},{"key":"ref_7","unstructured":"Bowers, Newton L., Gerber, Hans U., Hickman, James C., Jones, Donald A., and Nesbitt, Cecil J. (1986). Actuarial Mathematics, Society of Actuaries. [2nd ed.]."},{"key":"ref_8","unstructured":"B\u00fchlmann, Hans (1970). Mathematical Methods in Risk Theory, Springer."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"172","DOI":"10.1002\/asmb.1958","article-title":"Dividends in finite time horizon","volume":"30","author":"Cardoso","year":"2014","journal-title":"Applied Stochastic Models in Business and Industry"},{"key":"ref_10","first-page":"219","article-title":"Recursive calculation of time to ruin distributions","volume":"30","author":"Cardoso","year":"2002","journal-title":"Insurance: Mathematics and Economics"},{"key":"ref_11","first-page":"659","article-title":"Recursive calculation of finite time ruin probabilities under interest force","volume":"33","author":"Cardoso","year":"2003","journal-title":"Insurance: Mathematics and Economics"},{"key":"ref_12","first-page":"197","article-title":"Calculation of finite time ruin probabilities for some risk models","volume":"37","author":"Cardoso","year":"2005","journal-title":"Insurance: Mathematics and Economics"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1016\/j.ejor.2016.09.018","article-title":"On the dual risk model with parisian implementation delays in dividend payments","volume":"257","author":"Cheung","year":"2017","journal-title":"European Journal of Operational Research"},{"key":"ref_14","first-page":"84","article-title":"Cumulative parisian ruin in finite and infinite time horizons for a renewal risk process with exponential claims","volume":"111","author":"Cheung","year":"2023","journal-title":"Insurance: Mathematics and Economics"},{"key":"ref_15","unstructured":"Cram\u00e9r, Harald (1955). Collective Risk Theory: A Survey of the Theory from the Point of View of the Theory of Stochastic Processes, Nordiska Bokhandeln."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"54","DOI":"10.1016\/j.spl.2016.02.018","article-title":"A note on parisian ruin with an ultimate bankruptcy level for L\u00e9vy insurance risk processes","volume":"113","author":"Czarna","year":"2016","journal-title":"Statistics & Probability Letters"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"984","DOI":"10.1239\/jap\/1324046014","article-title":"Ruin probability with parisian delay for a spectrally negative L\u00e9vy risk process","volume":"48","author":"Czarna","year":"2011","journal-title":"Journal of Applied Probability"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1007\/s10957-013-0283-y","article-title":"Dividend problem with parisian delay for a spectrally negative L\u00e9vy risk process","volume":"161","author":"Czarna","year":"2014","journal-title":"Journal of Optimization Theory and Applications"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"499","DOI":"10.1016\/j.cam.2016.09.045","article-title":"The joint distribution of the parisian ruin time and the number of claims until parisian ruin in the classical risk model","volume":"313","author":"Czarna","year":"2017","journal-title":"Journal of Computational and Applied Mathematics"},{"key":"ref_20","first-page":"1","article-title":"Recursive calculation of finite-time ruin probabilities","volume":"7","author":"Goovaerts","year":"1988","journal-title":"Insurance: Mathematics and Economics"},{"key":"ref_21","unstructured":"Dickson, David C. M. (2016). Insurance Risk and Ruin, Cambridge University Press. [2nd ed.]. International Series on Actuarial Science."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"153","DOI":"10.2143\/AST.25.2.563245","article-title":"Some stable algorithms in ruin theory and their applications","volume":"25","author":"Dickson","year":"1995","journal-title":"ASTIN Bulletin"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"199","DOI":"10.2143\/AST.21.2.2005364","article-title":"Recursive calculation of survival probabilities","volume":"21","author":"Dickson","year":"1991","journal-title":"ASTIN Bulletin"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"49","DOI":"10.2143\/AST.34.1.504954","article-title":"Some optimal dividends problems","volume":"34","author":"Dickson","year":"2004","journal-title":"ASTIN Bulletin"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1080\/03461238.1984.10413758","article-title":"Approximations to ruin probability in the presence of an upper absorbing barrier","volume":"1984","author":"Dickson","year":"1984","journal-title":"Scandinavian Actuarial Journal"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"134","DOI":"10.1016\/j.ejor.2014.10.007","article-title":"On finite-time ruin probabilities in a generalized dual risk model with dependence","volume":"242","author":"Dimitrova","year":"2015","journal-title":"European Journal of Operational Research"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1002\/asmb.690","article-title":"On a compounding assets model with positive jumps","volume":"24","author":"Dong","year":"2008","journal-title":"Applied Stochastic Models in Business and Industry"},{"key":"ref_28","unstructured":"Gerber, Hans U. (1979). An Introduction to Mathematical Risk Theory, S.S. Huebner Foundation for Insurance Education."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Kaas, Rob, Goovaerts, Marc, Dhaene, Jan, and Denuit, Michel (2008). Modern Actuarial Risk Theory\u2014Using R, Springer. [2nd ed.].","DOI":"10.1007\/978-3-540-70998-5"},{"key":"ref_30","unstructured":"Klugman, Stuart A., Panjer, Harry H., and Willmot, Gordon E. (2019). Loss Models: From Data to Decisions, Wiley. Wiley Series in Probability and Statistics."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"599","DOI":"10.3150\/11-BEJ404","article-title":"Parisian ruin probability for spectrally negative L\u00e9vy processes","volume":"19","author":"Loeffen","year":"2013","journal-title":"Bernoulli"},{"key":"ref_32","first-page":"205","article-title":"A link between wave governed random motions and ruin processes","volume":"35","author":"Mazza","year":"2004","journal-title":"Insurance: Mathematics and Economics"},{"key":"ref_33","first-page":"429","article-title":"Self-similar processes in collective risk theory","volume":"11","author":"Michna","year":"1988","journal-title":"Journal of Applied Mathematics and Stochastic Analysis"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"22","DOI":"10.1017\/S0515036100006796","article-title":"Recursive evaluation of a family of compound distributions","volume":"12","author":"Panjer","year":"1981","journal-title":"ASTIN Bulletin"},{"key":"ref_35","first-page":"159","article-title":"Practical aspects of stop-loss calculations","volume":"2","author":"Panjer","year":"1983","journal-title":"Insurance: Mathematics and Economics"},{"key":"ref_36","first-page":"113","article-title":"Computational aspects of recursive evaluation of compound distributions","volume":"5","author":"Panjer","year":"1986","journal-title":"Insurance: Mathematics and Economics"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"227","DOI":"10.2143\/AST.23.2.2005093","article-title":"On the stability of recursive formulas","volume":"23","author":"Panjer","year":"1993","journal-title":"ASTIN Bulletin"},{"key":"ref_38","unstructured":"Rolski, Tomasz, Schmidli, Hanspeter, Schmidt, Volker, and Teugels, Jozef L. (2009). Stochastic Processes for Insurance and Finance, Wiley. Wiley Series in Probability and Statistics."},{"key":"ref_39","unstructured":"Seal, Hilary L. (1969). Stochastic Theory of a Risk Business, Wiley."},{"key":"ref_40","unstructured":"Tak\u00e1cs, Lajos (1967). Combinatorial Methods in the Theory of Stochastic Processes, Wiley."},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Xie, Jiayi, and Zhang, Zhimin (2025). Finite-time expected present value of operating costs until ruin in L\u00e9vy risk models with varying dividend barriers. Communications in Statistics\u2014Theory and Methods, 1\u201321.","DOI":"10.1080\/03610926.2025.2485343"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1080\/15326349.2014.868736","article-title":"The discounted moments of the surplus after the last innovation before ruin under the dual risk model","volume":"30","author":"Yang","year":"2014","journal-title":"Stochastic Models"},{"key":"ref_43","first-page":"135","article-title":"Parisian ruin with a threshold dividend strategy under the dual L\u00e9vy risk model","volume":"90","author":"Yang","year":"2020","journal-title":"Insurance: Mathematics and Economics"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"3298","DOI":"10.1080\/03610920802117080","article-title":"Ruin probabilities of a dual markov-modulated risk model","volume":"37","author":"Zhu","year":"2008","journal-title":"Communications in Statistics\u2014Theory and Methods"}],"container-title":["Risks"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2227-9091\/13\/9\/174\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:43:58Z","timestamp":1760035438000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2227-9091\/13\/9\/174"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,9,11]]},"references-count":44,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2025,9]]}},"alternative-id":["risks13090174"],"URL":"https:\/\/doi.org\/10.3390\/risks13090174","relation":{},"ISSN":["2227-9091"],"issn-type":[{"type":"electronic","value":"2227-9091"}],"subject":[],"published":{"date-parts":[[2025,9,11]]}}}