{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T12:58:27Z","timestamp":1740142707459,"version":"3.37.3"},"reference-count":0,"publisher":"Informa UK Limited","issue":"3","license":[{"start":{"date-parts":[[2003,1,1]],"date-time":"2003-01-01T00:00:00Z","timestamp":1041379200000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"China Bridges International"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Applied Mathematics and Decision Sciences"],"published-print":{"date-parts":[[2003,1,1]]},"abstract":"<jats:p>The computation of ruin probability is an important problem in the collective\nrisk theory. It has applications in the fields of insurance, actuarial science, and\neconomics. Many mathematical models have been introduced to simulate business activities\nand ruin probability is studied based on these models. Two of these models\nare the classical risk model and the Cox model. In the classical model, the counting\nprocess is a Poisson process and in the Cox model, the counting process is a Cox\nprocess. Thorin (1973) studied the ruin probability based on the classical model with\nthe assumption that random sequence followed the <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"$\\Gamma$\" id=\"E1\"><mml:mi>\u0393<\/mml:mi><\/mml:math> distribution with density function <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"$f(x)=\\frac{{x^{\\frac{1}{\\beta }- 1}}}{{\\beta ^{\\frac{1}{\\beta }} \\Gamma({1\/\\beta })}}e^{-\\frac{x}{\\beta}}$\" id=\"E2\"><mml:mi>f<\/mml:mi><mml:mrow><mml:mo>(<\/mml:mo><mml:mi>x<\/mml:mi><mml:mo>)<\/mml:mo><\/mml:mrow><mml:mo>=<\/mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mi>x<\/mml:mi><mml:mrow><mml:mfrac><mml:mn>1<\/mml:mn><mml:mi>\u03b2<\/mml:mi><\/mml:mfrac><mml:mo>\u2212<\/mml:mo><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msup><\/mml:mrow><mml:mrow><mml:msup><mml:mi>\u03b2<\/mml:mi><mml:mrow><mml:mfrac><mml:mn>1<\/mml:mn><mml:mi>\u03b2<\/mml:mi><\/mml:mfrac><\/mml:mrow><\/mml:msup><mml:mi>\u0393<\/mml:mi><mml:mrow><mml:mo>(<\/mml:mo><mml:mrow><mml:mn>1<\/mml:mn><mml:mo>\/<\/mml:mo><mml:mi>\u03b2<\/mml:mi><\/mml:mrow><mml:mo>)<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:mfrac><mml:msup><mml:mi>e<\/mml:mi><mml:mrow><mml:mo>\u2212<\/mml:mo><mml:mfrac><mml:mi>x<\/mml:mi><mml:mi>\u03b2<\/mml:mi><\/mml:mfrac><\/mml:mrow><\/mml:msup><\/mml:math>, <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"$x&gt;0$\" id=\"E3\"><mml:mi>x<\/mml:mi><mml:mo>&gt;<\/mml:mo><mml:mn>0<\/mml:mn><\/mml:math>, where <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"$\\beta&gt;1$\" id=\"E4\"><mml:mi>\u03b2<\/mml:mi><mml:mo>&gt;<\/mml:mo><mml:mn>1<\/mml:mn><\/mml:math>. This paper studies the ruin probability of the classical model where the random sequence follows the <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"$\\Gamma$\" id=\"E5\"><mml:mi>\u0393<\/mml:mi><\/mml:math> distribution with density function <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"$f(x)=\\frac{{\\alpha^n}}{{\\Gamma(n)}}x^{n-1} e^{-\\alpha x}$\" id=\"E6\"><mml:mi>f<\/mml:mi><mml:mrow><mml:mo>(<\/mml:mo><mml:mi>x<\/mml:mi><mml:mo>)<\/mml:mo><\/mml:mrow><mml:mo>=<\/mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mi>\u03b1<\/mml:mi><mml:mi>n<\/mml:mi><\/mml:msup><\/mml:mrow><mml:mrow><mml:mi>\u0393<\/mml:mi><mml:mrow><mml:mo>(<\/mml:mo><mml:mi>n<\/mml:mi><mml:mo>)<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:mfrac><mml:msup><mml:mi>x<\/mml:mi><mml:mrow><mml:mi>n<\/mml:mi><mml:mo>\u2212<\/mml:mo><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msup><mml:msup><mml:mi>e<\/mml:mi><mml:mrow><mml:mo>\u2212<\/mml:mo><mml:mi>\u03b1<\/mml:mi><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:msup><\/mml:math>, <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"$x&gt;0$\" id=\"E7\"><mml:mi>x<\/mml:mi><mml:mo>&gt;<\/mml:mo><mml:mn>0<\/mml:mn><\/mml:math>, where <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"$\\alpha&gt;0$\" id=\"E8\"><mml:mi>\u03b1<\/mml:mi><mml:mo>&gt;<\/mml:mo><mml:mn>0<\/mml:mn><\/mml:math> and <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"$n \\geqslant 2$\" id=\"E9\"><mml:mi>n<\/mml:mi><mml:mo>\u2265<\/mml:mo><mml:mn>2<\/mml:mn><\/mml:math> is a positive integer. An intermediate general result is given and a complete solution is provided for <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"$n=2$\" id=\"E10\"><mml:mi>n<\/mml:mi><mml:mo>=<\/mml:mo><mml:mn>2<\/mml:mn><\/mml:math>. Simulation studies for the case of <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"$n=2$\" id=\"E11\"><mml:mi>n<\/mml:mi><mml:mo>=<\/mml:mo><mml:mn>2<\/mml:mn><\/mml:math> is also provided.<\/jats:p>","DOI":"10.1155\/s1173912603000130","type":"journal-article","created":{"date-parts":[[2007,3,8]],"date-time":"2007-03-08T07:44:28Z","timestamp":1173339868000},"page":"133-146","source":"Crossref","is-referenced-by-count":0,"title":["Some results of ruin probability for the classical risk process"],"prefix":"10.1080","volume":"7","author":[{"given":"He","family":"Yuanjiang","sequence":"first","affiliation":[{"name":"Department of Statistics Science, Zhongshan University, Ganglion, 510275 PR, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Li","family":"Xucheng","sequence":"additional","affiliation":[{"name":"Department of Statistics Science, Zhongshan University, Ganglion, 510275 PR, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"John","family":"Zhang","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Indiana University of Pennsylvania, Indiana, PA 15705, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"301","container-title":["Journal of Applied Mathematics and Decision Sciences"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/archive\/2003\/526402.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/archive\/2003\/526402.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,8,8]],"date-time":"2024-08-08T14:23:14Z","timestamp":1723126994000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.hindawi.com\/journals\/ads\/2003\/526402\/abs\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2003,1,1]]},"references-count":0,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2003,1,1]]}},"alternative-id":["526402"],"URL":"https:\/\/doi.org\/10.1155\/s1173912603000130","relation":{},"ISSN":["1173-9126"],"issn-type":[{"type":"print","value":"1173-9126"}],"subject":[],"published":{"date-parts":[[2003,1,1]]}}}