{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:47:06Z","timestamp":1760060826307,"version":"build-2065373602"},"reference-count":33,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2025,9,23]],"date-time":"2025-09-23T00:00:00Z","timestamp":1758585600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["11901398","62473157","62333006","202201010250"],"award-info":[{"award-number":["11901398","62473157","62333006","202201010250"]}]},{"name":"Basic and Applied Basic Research of Guangzhou Basic Research Program","award":["11901398","62473157","62333006","202201010250"],"award-info":[{"award-number":["11901398","62473157","62333006","202201010250"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This study focuses on analyzing the almost sure exponential stability of the split-step \u03b8-method (SS\u03b8-method) when applied to stochastic pantograph differential equations characterized by Markovian switching and jump processes. Initially, we establish the almost sure exponential stability of the system\u2019s trivial solution. Subsequently, under an additional sufficient condition, it is demonstrated that the discrete solutions generated by the SS\u03b8-method also exhibit this stability property. Finally, a computational experiment is conducted to support the theoretical results.<\/jats:p>","DOI":"10.3390\/axioms14100718","type":"journal-article","created":{"date-parts":[[2025,9,23]],"date-time":"2025-09-23T13:48:37Z","timestamp":1758635317000},"page":"718","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Stability of the Split-Step \u03b8-Method for Stochastic Pantograph Systems with Markovian Switching and Jumps"],"prefix":"10.3390","volume":"14","author":[{"given":"Guangjie","family":"Li","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Guangdong University of Foreign Studies, Guangzhou 510006, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhipei","family":"Hu","sequence":"additional","affiliation":[{"name":"School of Automation Science and Engineering, South China University of Technology, Guangzhou 510641, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Baishu","family":"Xu","sequence":"additional","affiliation":[{"name":"School of Automation Science and Engineering, South China University of Technology, Guangzhou 510641, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zilong","family":"Chen","sequence":"additional","affiliation":[{"name":"School of Automation, Guang Dong Polytechnic Normal University, Guangzhou 510665, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Feiqi","family":"Deng","sequence":"additional","affiliation":[{"name":"School of Automation Science and Engineering, South China University of Technology, Guangzhou 510641, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,9,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1142\/S0219493705001353","article-title":"Decay and growth rates of solutions of scalar stochastic delay differential equations with unbounded delay and state dependent noise","volume":"5","author":"Appleby","year":"2005","journal-title":"Stoch. 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