{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,23]],"date-time":"2026-08-23T14:48:23Z","timestamp":1787496503495,"version":"build-2736575974"},"reference-count":8,"publisher":"Oxford University Press (OUP)","issue":"2","license":[{"start":{"date-parts":[[1960,7,1]],"date-time":"1960-07-01T00:00:00Z","timestamp":-299894400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[1960,7,1]]},"abstract":"<jats:title>SUMMARY<\/jats:title>\n                  <jats:p>A study is made of approximations of Green's type (often called W.K.B. approximations) to the solution of equations governing univariate stochastic processes. After considering some specific processes, the general homogeneous process in continuous time is discussed. A numerical comparison is made between the approximate and exact probabilities for the multiplicative birth and death process. Diffusion processes are briefly examined. The method is applied to the equations both for probabilities and for the moment generating function, and the duality between the approximate solutions is indicated. Finally, approximate formulae for the cumulants are obtained and the limitations of the method are pointed out.<\/jats:p>","DOI":"10.1111\/j.2517-6161.1960.tb00385.x","type":"journal-article","created":{"date-parts":[[2018,12,7]],"date-time":"2018-12-07T01:43:38Z","timestamp":1544147018000},"page":"376-401","source":"Crossref","is-referenced-by-count":18,"title":["Approximate Solutions of Green's Type for Univariate Stochastic Processes"],"prefix":"10.1093","volume":"22","author":[{"given":"H. E.","family":"Daniels","sequence":"first","affiliation":[{"name":"University of Birmingham"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2018,12,5]]},"reference":[{"key":"2024090921201088800_cit1","volume-title":"Stochastic Processes","author":"Bartlett","year":"1955"},{"key":"2024090921201088800_cit2","first-page":"1","article-title":"Some genetical problems in natural populations","volume":"4","author":"Crow","year":"1956","journal-title":"Proc. Third Berkeley Symposium"},{"key":"2024090921201088800_cit3","doi-asserted-by":"crossref","first-page":"631","DOI":"10.1214\/aoms\/1177728652","article-title":"Saddle point approximations in statistics","volume":"25","author":"Daniels","year":"1954","journal-title":"Ann. Math. Statist."},{"key":"2024090921201088800_cit4","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1525\/9780520411586-018","article-title":"Diffusion processes in genetics","volume-title":"Proc. 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