{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,19]],"date-time":"2026-02-19T04:47:43Z","timestamp":1771476463479,"version":"3.50.1"},"reference-count":21,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2024,3,6]],"date-time":"2024-03-06T00:00:00Z","timestamp":1709683200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Natural Science Foundation of the Shandong Province of China","award":["ZR202111290596"],"award-info":[{"award-number":["ZR202111290596"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>For a family of stochastic differential equations driven by additive Gaussian noise, we study the asymptotic behaviors of its corresponding Euler\u2013Maruyama scheme by deriving its convergence rate in terms of relative entropy. Our results for the convergence rate in terms of relative entropy complement the conventional ones in the strong and weak sense and induce some other properties of the Euler\u2013Maruyama scheme. For example, the convergence in terms of the total variation distance can be implied by Pinsker\u2019s inequality directly. Moreover, when the drift is \u03b2(0&lt;\u03b2&lt;1)-H\u00f6lder continuous in the spatial variable, the convergence rate in terms of the weighted variation distance is also established. Both of these convergence results do not seem to be directly obtained from any other convergence results of the Euler\u2013Maruyama scheme. The main tool this paper relies on is the Girsanov transform.<\/jats:p>","DOI":"10.3390\/e26030232","type":"journal-article","created":{"date-parts":[[2024,3,6]],"date-time":"2024-03-06T04:03:50Z","timestamp":1709697830000},"page":"232","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Convergence of Relative Entropy for Euler\u2013Maruyama Scheme to Stochastic Differential Equations with Additive Noise"],"prefix":"10.3390","volume":"26","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6415-6843","authenticated-orcid":false,"given":"Yuan","family":"Yu","sequence":"first","affiliation":[{"name":"School of Statistics and Mathematics, Shandong University of Finance and Economics, Jinan 250014, China"}]}],"member":"1968","published-online":{"date-parts":[[2024,3,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Kloden, P.E., and Platen, E. (1992). 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