{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,19]],"date-time":"2026-03-19T12:48:37Z","timestamp":1773924517637,"version":"3.50.1"},"reference-count":28,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2012,9,10]],"date-time":"2012-09-10T00:00:00Z","timestamp":1347235200000},"content-version":"vor","delay-in-days":253,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2012,1]]},"abstract":"<jats:p>We have presented a new unified approach to model the dynamics of both the sum and difference of two correlated lognormal stochastic variables. By the Lie\u2010Trotter operator splitting method, both the sum and difference are shown to follow a shifted lognormal stochastic process, and approximate probability distributions are determined in closed form. Illustrative numerical examples are presented to demonstrate the validity and accuracy of these approximate distributions. In terms of the approximate probability distributions, we have also obtained an analytical series expansion of the exact solutions, which can allow us to improve the approximation in a systematic manner. Moreover, we believe that this new approach can be extended to study both (1) the algebraic sum of <jats:italic>N<\/jats:italic> lognormals, and (2) the sum and difference of other correlated stochastic processes, for example, two correlated CEV processes, two correlated CIR processes, and two correlated lognormal processes with mean\u2010reversion.<\/jats:p>","DOI":"10.1155\/2012\/838397","type":"journal-article","created":{"date-parts":[[2012,9,10]],"date-time":"2012-09-10T21:02:27Z","timestamp":1347310947000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":29,"title":["The Sum and Difference of Two Lognormal Random Variables"],"prefix":"10.1155","volume":"2012","author":[{"given":"C. F.","family":"Lo","sequence":"first","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2012,9,10]]},"reference":[{"key":"e_1_2_6_1_2","doi-asserted-by":"publisher","DOI":"10.1109\/TCOM.1960.1097606"},{"key":"e_1_2_6_2_2","doi-asserted-by":"publisher","DOI":"10.1080\/01621459.1969.10501004"},{"key":"e_1_2_6_3_2","doi-asserted-by":"publisher","DOI":"10.1080\/01621459.1971.10482229"},{"key":"e_1_2_6_4_2","doi-asserted-by":"publisher","DOI":"10.1109\/25.467959"},{"key":"e_1_2_6_5_2","doi-asserted-by":"crossref","unstructured":"WuJ. MehtaN. B. andZhangJ. Flexible lognormal sum approximation method 6 Proceedings of IEEE Global Telecommunications Conference (GLOBECOM \u203205) December 2005 St. Louis Mo USA 3413\u20133417 2-s2.0-33846563530 https:\/\/doi.org\/10.1109\/GLOCOM.2005.1578407.","DOI":"10.1109\/GLOCOM.2005.1578407"},{"key":"e_1_2_6_6_2","doi-asserted-by":"publisher","DOI":"10.1155\/2009\/630857"},{"key":"e_1_2_6_7_2","doi-asserted-by":"publisher","DOI":"10.2307\/2331102"},{"key":"e_1_2_6_8_2","doi-asserted-by":"publisher","DOI":"10.1137\/S0036144503424798"},{"key":"e_1_2_6_9_2","doi-asserted-by":"publisher","DOI":"10.1239\/aap\/1093962232"},{"key":"e_1_2_6_10_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0167-6687(02)00135-X"},{"key":"e_1_2_6_11_2","doi-asserted-by":"publisher","DOI":"10.1080\/10920277.2005.10596226"},{"key":"e_1_2_6_12_2","first-page":"82","article-title":"Bounds for a sum of random variables under a mixture of normals","volume":"13","author":"Kukush A.","year":"2007","journal-title":"Theory of Stochastic Processes"},{"key":"e_1_2_6_13_2","doi-asserted-by":"publisher","DOI":"10.1046\/j.1095-8312.2003.00220.x"},{"key":"e_1_2_6_14_2","doi-asserted-by":"publisher","DOI":"10.1140\/epjb\/e2003-00131-6"},{"key":"e_1_2_6_15_2","volume-title":"Lognormal Distributions: Theory and Applications","author":"Crow E. L.","year":"1988"},{"key":"e_1_2_6_16_2","doi-asserted-by":"publisher","DOI":"10.1641\/0006-3568(2001)051[0341:LNDATS]2.0.CO;2"},{"key":"e_1_2_6_17_2","doi-asserted-by":"publisher","DOI":"10.1109\/LCOMM.2004.837657"},{"key":"e_1_2_6_18_2","doi-asserted-by":"publisher","DOI":"10.1109\/TVT.2007.905611"},{"key":"e_1_2_6_19_2","doi-asserted-by":"publisher","DOI":"10.1109\/LCOMM.2006.1613731"},{"key":"e_1_2_6_20_2","doi-asserted-by":"publisher","DOI":"10.1109\/TVT.2007.891467"},{"key":"e_1_2_6_21_2","doi-asserted-by":"publisher","DOI":"10.1109\/TWC.2007.051000"},{"key":"e_1_2_6_22_2","doi-asserted-by":"publisher","DOI":"10.3905\/jod.2007.686420"},{"key":"e_1_2_6_23_2","unstructured":"DufresneD. Sums of lognormals Proceedings of the 43rd Actuarial Research Conference August 2008 Regina Canada University of Regina."},{"key":"e_1_2_6_24_2","doi-asserted-by":"publisher","DOI":"10.1137\/090750421"},{"key":"e_1_2_6_25_2","doi-asserted-by":"publisher","DOI":"10.1109\/TVT.2011.2163652"},{"key":"e_1_2_6_26_2","doi-asserted-by":"publisher","DOI":"10.1109\/TCOMM.2011.091911.110054"},{"key":"e_1_2_6_27_2","doi-asserted-by":"publisher","DOI":"10.3905\/jod.2012.19.3.077"},{"key":"e_1_2_6_28_2","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1959-0108732-6"}],"container-title":["Journal of Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2012\/838397.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2012\/838397.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1155\/2012\/838397","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,6,11]],"date-time":"2024-06-11T07:05:51Z","timestamp":1718089551000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1155\/2012\/838397"}},"subtitle":[],"editor":[{"given":"Mehmet","family":"Sezer","sequence":"additional","affiliation":[]}],"short-title":[],"issued":{"date-parts":[[2012,1]]},"references-count":28,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2012,1]]}},"alternative-id":["10.1155\/2012\/838397"],"URL":"https:\/\/doi.org\/10.1155\/2012\/838397","archive":["Portico"],"relation":{},"ISSN":["1110-757X","1687-0042"],"issn-type":[{"value":"1110-757X","type":"print"},{"value":"1687-0042","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,1]]},"assertion":[{"value":"2012-05-15","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2012-07-19","order":1,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2012-09-10","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}],"article-number":"838397"}}