{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,14]],"date-time":"2026-01-14T17:02:15Z","timestamp":1768410135714,"version":"3.49.0"},"reference-count":16,"publisher":"Wiley","license":[{"start":{"date-parts":[[2012,4,1]],"date-time":"2012-04-01T00:00:00Z","timestamp":1333238400000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["LMS J. Comput. Math."],"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We perform an almost sure\u00a0linear stability analysis of the <jats:italic>\u03b8<\/jats:italic>-Maruyama method, selecting as our test equation a two-dimensional system of It\u00f4 differential equations with diagonal drift coefficient and two independent stochastic perturbations which capture the stabilising and destabilising roles of feedback geometry in the almost sure\u00a0asymptotic stability of the equilibrium solution. For small values of the constant step-size parameter, we derive close-to-sharp conditions for the almost sure\u00a0asymptotic stability and instability of the equilibrium solution of the discretisation that match those of the original test system. Our investigation demonstrates the use of a discrete form of the It\u00f4 formula in the context of an almost sure\u00a0linear stability analysis.<\/jats:p>","DOI":"10.1112\/s1461157012000010","type":"journal-article","created":{"date-parts":[[2012,4,5]],"date-time":"2012-04-05T11:07:07Z","timestamp":1333624027000},"page":"71-83","source":"Crossref","is-referenced-by-count":18,"title":["Almost sure asymptotic stability analysis of the <i>\u03b8<\/i>-Maruyama method applied to a test system with stabilising and destabilising stochastic perturbations"],"prefix":"10.1112","volume":"15","author":[{"given":"Gregory","family":"Berkolaiko","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Evelyn","family":"Buckwar","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"C\u00f3nall","family":"Kelly","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Alexandra","family":"Rodkina","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2012,4,1]]},"reference":[{"key":"S1461157012000010_ref1","first-page":"202","article-title":"Discrete It\u00f4 formulas and their applications to stochastic numerics","volume":"1462","author":"Akahori","year":"2006","journal-title":"RIMS K\u00f4ky\u00fbroku Bessatsu"},{"key":"S1461157012000010_ref4","first-page":"614","volume-title":"Proceedings of the International Conference 2004 \u2013 Dynamical Systems and Applications, Antalya, Turkey, 5\u201310 July 2004","author":"Berkolaiko"},{"key":"S1461157012000010_ref15","volume-title":"Stochastic differential equations and their applications","author":"Mao","year":"1997"},{"key":"S1461157012000010_ref2","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1080\/17442500802088541","article-title":"Non-exponential stability and decay rates in nonlinear stochastic difference equations with unbounded noises","volume":"81","author":"Appleby","year":"2009","journal-title":"Stochastics"},{"key":"S1461157012000010_ref3","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.2008.919255"},{"key":"S1461157012000010_ref7","doi-asserted-by":"publisher","DOI":"10.1214\/aoms\/1177705909"},{"key":"S1461157012000010_ref12","doi-asserted-by":"publisher","DOI":"10.1137\/060658138"},{"key":"S1461157012000010_ref11","doi-asserted-by":"publisher","DOI":"10.1137\/S003614299834736X"},{"key":"S1461157012000010_ref16","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-2539-1"},{"key":"S1461157012000010_ref5","doi-asserted-by":"publisher","DOI":"10.1137\/090771843"},{"key":"S1461157012000010_ref8","volume-title":"Probability and random processes","author":"Grimmett","year":"1991"},{"key":"S1461157012000010_ref13","doi-asserted-by":"publisher","DOI":"10.1081\/SAP-120014691"},{"key":"S1461157012000010_ref9","first-page":"3","article-title":"An expos\u00e9 on discrete Wiener chaos expansions","volume":"XIII","author":"Gzyl","year":"2006","journal-title":"Bol. 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