{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,10]],"date-time":"2026-07-10T03:02:26Z","timestamp":1783652546705,"version":"3.55.0"},"reference-count":30,"publisher":"Walter de Gruyter GmbH","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2016,12,1]]},"abstract":"Abstract<\/jats:title>\n In this paper, we consider both, the strong and weak convergence of the Euler\u2013Maruyama approximation for one-dimensional stochastic differential equations involving the local times of the unknown process. We use a transformation in order to remove the local time \n \n \n \n L<\/m:mi>\n t<\/m:mi>\n a<\/m:mi>\n <\/m:msubsup>\n <\/m:math>\n \n ${L_{t}^{a}}$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> from the stochastic differential equations of type<\/jats:p>\n \n \n \n \n \n \n \n X<\/m:mi>\n t<\/m:mi>\n <\/m:msub>\n =<\/m:mo>\n \n \n X<\/m:mi>\n 0<\/m:mn>\n <\/m:msub>\n +<\/m:mo>\n \n \n \u222b<\/m:mo>\n 0<\/m:mn>\n t<\/m:mi>\n <\/m:msubsup>\n \n \u03c6<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \n X<\/m:mi>\n s<\/m:mi>\n <\/m:msub>\n )<\/m:mo>\n <\/m:mrow>\n \u2062<\/m:mo>\n \n \ud835\udc51<\/m:mo>\n \n B<\/m:mi>\n s<\/m:mi>\n <\/m:msub>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:mrow>\n +<\/m:mo>\n \n \n \u222b<\/m:mo>\n \u211d<\/m:mi>\n <\/m:msub>\n \n \u03bd<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \n d<\/m:mi>\n \u2062<\/m:mo>\n a<\/m:mi>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n \u2062<\/m:mo>\n \n L<\/m:mi>\n t<\/m:mi>\n a<\/m:mi>\n <\/m:msubsup>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:mrow>\n .<\/m:mo>\n <\/m:mrow>\n <\/m:math>\n \n $X_{t}=X_{0}+\\int_{0}^{t}\\varphi(X_{s})\\,dB_{s}+\\int_{\\mathbb{R}}\\nu(da)L_{t}^{%\na}.$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:disp-formula>\n <\/jats:p>\n Here B<\/jats:italic> is a one-dimensional Brownian motion, \n \n \n \n \u03c6<\/m:mi>\n :<\/m:mo>\n \n \u211d<\/m:mi>\n \u2192<\/m:mo>\n \u211d<\/m:mi>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n ${\\varphi:\\mathbb{R}\\rightarrow\\mathbb{R}}$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> is a bounded measurable function, and \u03bd is a bounded measure on \n \n \n \u211d<\/m:mi>\n <\/m:math>\n \n ${\\mathbb{R}}$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>. We provide the approximation of Euler\u2013Maruyama for the stochastic differential equations without local time. After that, we conclude the approximation of Euler\u2013Maruyama \n \n \n \n X<\/m:mi>\n t<\/m:mi>\n n<\/m:mi>\n <\/m:msubsup>\n <\/m:math>\n \n ${X_{t}^{n}}$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> of the above mentioned equation, and we provide the rate of strong convergence \n \n \n \n Error<\/m:mo>\n =<\/m:mo>\n \n \ud835\udd3c<\/m:mi>\n \u2062<\/m:mo>\n \n |<\/m:mo>\n \n \n X<\/m:mi>\n T<\/m:mi>\n <\/m:msub>\n -<\/m:mo>\n \n X<\/m:mi>\n T<\/m:mi>\n n<\/m:mi>\n <\/m:msubsup>\n <\/m:mrow>\n |<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n ${\\operatorname{Error}=\\mathbb{E}\\lvert X_{T}-X_{T}^{n}\\rvert}$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, and the rate of weak convergence \n \n \n \n Error<\/m:mo>\n =<\/m:mo>\n \n \ud835\udd3c<\/m:mi>\n \u2062<\/m:mo>\n \n |<\/m:mo>\n \n \n G<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \n X<\/m:mi>\n T<\/m:mi>\n <\/m:msub>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n -<\/m:mo>\n \n G<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \n X<\/m:mi>\n T<\/m:mi>\n n<\/m:mi>\n <\/m:msubsup>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:mrow>\n |<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n ${\\operatorname{Error}=\\mathbb{E}\\lvert G(X_{T})-G(X_{T}^{n})\\rvert}$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, for any function \n \n \n \n G<\/m:mi>\n :<\/m:mo>\n \n \u211d<\/m:mi>\n \u2192<\/m:mo>\n \u211d<\/m:mi>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n ${G:\\mathbb{R}\\rightarrow\\mathbb{R}}$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> of bounded variation.<\/jats:p>","DOI":"10.1515\/mcma-2016-0115","type":"journal-article","created":{"date-parts":[[2016,10,26]],"date-time":"2016-10-26T10:01:13Z","timestamp":1477476073000},"page":"307-322","source":"Crossref","is-referenced-by-count":8,"title":["Approximation of Euler\u2013Maruyama for one-dimensional stochastic differential equations involving the local times of the unknown process"],"prefix":"10.1515","volume":"22","author":[{"given":"Mohsine","family":"Benabdallah","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Ibn Tofail, 14000 Kenitra, Morocco"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Youssfi","family":"Elkettani","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Ibn Tofail, 14000 Kenitra, Morocco"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Kamal","family":"Hiderah","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Ibn Tofail, 14000 Kenitra, Morocco"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"374","published-online":{"date-parts":[[2016,10,25]]},"reference":[{"key":"2023040102000373829_j_mcma-2016-0115_ref_001_w2aab2b8d829b1b7b1ab2ab1Aa","doi-asserted-by":"crossref","unstructured":"Avikainen R.,\nOn irregular functionals of SDEs and the Euler scheme,\nFinance Stoch. 13 (2009), 381\u2013401.","DOI":"10.1007\/s00780-009-0099-7"},{"key":"2023040102000373829_j_mcma-2016-0115_ref_002_w2aab2b8d829b1b7b1ab2ab2Aa","doi-asserted-by":"crossref","unstructured":"Bally V. and Talay D.,\nThe law of the Euler scheme for stochastic differential equations. 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