{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,13]],"date-time":"2025-12-13T06:57:12Z","timestamp":1765609032481,"version":"3.40.5"},"reference-count":17,"publisher":"Walter de Gruyter GmbH","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2024,3,1]]},"abstract":"Abstract<\/jats:title>\n In this paper, under some suitable conditions, we prove existence of a strong solution and uniqueness for the perturbed stochastic differential equations with reflected boundary (PSDERB), that is,<\/jats:p>\n \n \n \n \n \n {<\/m:mo>\n \n \n \n \n x<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n t<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:mtd>\n \n \n \n \n =<\/m:mo>\n \n \n x<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n 0<\/m:mn>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n +<\/m:mo>\n \n \n \u222b<\/m:mo>\n 0<\/m:mn>\n t<\/m:mi>\n <\/m:msubsup>\n \n \u03c3<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n s<\/m:mi>\n ,<\/m:mo>\n \n x<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n s<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n \u2062<\/m:mo>\n \n d<\/m:mo>\n B<\/m:mi>\n <\/m:mrow>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n s<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:mrow>\n +<\/m:mo>\n \n \n \u222b<\/m:mo>\n 0<\/m:mn>\n t<\/m:mi>\n <\/m:msubsup>\n \n b<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n s<\/m:mi>\n ,<\/m:mo>\n \n x<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n s<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n \u2062<\/m:mo>\n \n d<\/m:mo>\n s<\/m:mi>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:mrow>\n +<\/m:mo>\n \n \u03b1<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n t<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n \u2062<\/m:mo>\n H<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \n \n \n max<\/m:mi>\n \n 0<\/m:mn>\n \u2264<\/m:mo>\n u<\/m:mi>\n \u2264<\/m:mo>\n t<\/m:mi>\n <\/m:mrow>\n <\/m:munder>\n \u2061<\/m:mo>\n x<\/m:mi>\n <\/m:mrow>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n u<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n +<\/m:mo>\n \n \u03b2<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n t<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n \u2062<\/m:mo>\n \n L<\/m:mi>\n t<\/m:mi>\n 0<\/m:mn>\n <\/m:msubsup>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n x<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:mrow>\n ,<\/m:mo>\n <\/m:mrow>\n <\/m:mtd>\n <\/m:mtr>\n \n \n \n x<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n t<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:mtd>\n \n \n \n \n \n \u2265<\/m:mo>\n 0<\/m:mn>\n <\/m:mrow>\n \n \n \n for all<\/m:mtext>\n \u2062<\/m:mo>\n t<\/m:mi>\n <\/m:mrow>\n \u2265<\/m:mo>\n 0<\/m:mn>\n <\/m:mrow>\n <\/m:mrow>\n ,<\/m:mo>\n <\/m:mrow>\n <\/m:mtd>\n <\/m:mtr>\n <\/m:mtable>\n <\/m:mrow>\n <\/m:math>\n \n \\left\\{\\begin{aligned} {}x(t)&=x(0)+\\int_{0}^{t}\\sigma(s,x(s))\\,dB(s)+\\int_{0}^{t}b(s,x(s))\\,ds+\\alpha(t)H\\bigl{(}\\max_{0\\leq u\\leq t}x(u)\\bigr{)}+\\beta(t)L_{t}^{0}(x),\\\\\nx(t)&\\geq 0\\quad\\text{for all}\\ t\\geq 0,\\end{aligned}\\right.<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:disp-formula>\n <\/jats:p>\n where \ud835\udc3b is a continuous R-valued function, \n \n \n \n \u03c3<\/m:mi>\n ,<\/m:mo>\n b<\/m:mi>\n ,<\/m:mo>\n \u03b1<\/m:mi>\n <\/m:mrow>\n <\/m:math>\n \n \\sigma,b,\\alpha<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> and \ud835\udefd are measurable functions, \n \n \n \n L<\/m:mi>\n t<\/m:mi>\n 0<\/m:mn>\n <\/m:msubsup>\n <\/m:math>\n \n L_{t}^{0}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> denotes a local time at point zero for the time of the semi-martingale \ud835\udc65.<\/jats:p>","DOI":"10.1515\/mcma-2023-2018","type":"journal-article","created":{"date-parts":[[2023,10,23]],"date-time":"2023-10-23T17:40:55Z","timestamp":1698082855000},"page":"31-41","source":"Crossref","is-referenced-by-count":2,"title":["Existence and uniqueness of solutions for perturbed stochastic differential equations with reflected boundary"],"prefix":"10.1515","volume":"30","author":[{"given":"Faiz","family":"Bahaj","sequence":"first","affiliation":[{"name":"Department of Mathematics , Faculty of Education-Ateq , Shabwh University , Shabwah , Yemen"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8165-8685","authenticated-orcid":false,"given":"Kamal","family":"Hiderah","sequence":"additional","affiliation":[{"name":"Department of Mathematics , Faculty of Science , Aden University , Aden , Yemen"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2023,10,24]]},"reference":[{"key":"2024022618254286523_j_mcma-2023-2018_ref_001","doi-asserted-by":"crossref","unstructured":"R. Belfadli, S. Hamad\u00e8ne and Y. Ouknine,\nOn one-dimensional stochastic differential equations involving the maximum process,\nStoch. Dyn. 9 (2009), no. 2, 277\u2013292.","DOI":"10.1142\/S0219493709002671"},{"key":"2024022618254286523_j_mcma-2023-2018_ref_002","doi-asserted-by":"crossref","unstructured":"M. Benabdallah and K. Hiderah,\nStrong rate of convergence for the Euler\u2013Maruyama approximation of one-dimensional stochastic differential equations involving the local time at point zero,\nMonte Carlo Methods Appl. 24 (2018), no. 4, 249\u2013262.","DOI":"10.1515\/mcma-2018-2021"},{"key":"2024022618254286523_j_mcma-2023-2018_ref_003","doi-asserted-by":"crossref","unstructured":"S. Bouhadou and Y. Ouknine,\nOn the time inhomogeneous skew Brownian motion,\nBull. Sci. Math. 137 (2013), no. 7, 835\u2013850.","DOI":"10.1016\/j.bulsci.2013.02.001"},{"key":"2024022618254286523_j_mcma-2023-2018_ref_004","doi-asserted-by":"crossref","unstructured":"P. Carmona, F. Petit and M. Yor,\nBeta variables as times spent in \n \n \n \n [\n 0\n ,\n \u221e\n [\n \n \n \n [0,\\infty[\n \n by certain perturbed Brownian motions,\nJ. Lond. Math. Soc. (2) 58 (1998), no. 1, 239\u2013256.","DOI":"10.1112\/S0024610798006401"},{"key":"2024022618254286523_j_mcma-2023-2018_ref_005","doi-asserted-by":"crossref","unstructured":"L. Chaumont and R. A. Doney,\nPathwise uniqueness for perturbed versions of Brownian motion and reflected Brownian motion,\nProbab. Theory Related Fields 113 (1999), no. 4, 519\u2013534.","DOI":"10.1007\/s004400050216"},{"key":"2024022618254286523_j_mcma-2023-2018_ref_006","doi-asserted-by":"crossref","unstructured":"L. Chaumont and R. A. Doney,\nSome calculations for doubly perturbed Brownian motion,\nStochastic Process. Appl. 85 (2000), no. 1, 61\u201374.","DOI":"10.1016\/S0304-4149(99)00065-4"},{"key":"2024022618254286523_j_mcma-2023-2018_ref_007","doi-asserted-by":"crossref","unstructured":"B. Davis,\nBrownian motion and random walk perturbed at extrema,\nProbab. Theory Related Fields 113 (1999), no. 4, 501\u2013518.","DOI":"10.1007\/s004400050215"},{"key":"2024022618254286523_j_mcma-2023-2018_ref_008","doi-asserted-by":"crossref","unstructured":"R. A. Doney,\nSome calculations for perturbed Brownian motion,\nS\u00e9minaire de Probabilit\u00e9s. XXXII,\nLecture Notes in Math. 1686,\nSpringer, Berlin (1998), 231\u2013236.","DOI":"10.1007\/BFb0101760"},{"key":"2024022618254286523_j_mcma-2023-2018_ref_009","doi-asserted-by":"crossref","unstructured":"R. A. Doney and T. Zhang,\nPerturbed Skorohod equations and perturbed reflected diffusion processes,\nAnn. Inst. H. Poincar\u00e9 Probab. Statist. 41 (2005), no. 1, 107\u2013121.","DOI":"10.1016\/j.anihpb.2004.03.005"},{"key":"2024022618254286523_j_mcma-2023-2018_ref_010","doi-asserted-by":"crossref","unstructured":"P. \u00c9tor\u00e9 and M. Martinez,\nOn the existence of a time inhomogeneous skew Brownian motion and some related laws,\nElectron. J. Probab. 17 (2012), 1\u201327.","DOI":"10.1214\/EJP.v17-1858"},{"key":"2024022618254286523_j_mcma-2023-2018_ref_011","doi-asserted-by":"crossref","unstructured":"P. \u00c9tor\u00e9 and M. Martinez,\nTime inhomogeneous stochastic differential equations involving the local time of the unknown process, and associated parabolic operators,\nStochastic Process. Appl. 128 (2018), no. 8, 2642\u20132687.","DOI":"10.1016\/j.spa.2017.09.018"},{"key":"2024022618254286523_j_mcma-2023-2018_ref_012","doi-asserted-by":"crossref","unstructured":"K. Hiderah,\nCarath\u00e9odory approximate solutions for a class of stochastic differential equations involving the local time at point zero with one-sided Lipschitz continuous drift coefficients,\nMonte Carlo Methods Appl. 28 (2022), no. 2, 189\u2013198.","DOI":"10.1515\/mcma-2022-2107"},{"key":"2024022618254286523_j_mcma-2023-2018_ref_013","doi-asserted-by":"crossref","unstructured":"K. Hiderah,\nExistence and pathwise uniqueness of solutions for stochastic differential equations involving the local time at point zero,\nStoch. Anal. Appl. 41 (2023), no. 2, 276\u2013290.","DOI":"10.1080\/07362994.2021.2011317"},{"key":"2024022618254286523_j_mcma-2023-2018_ref_014","doi-asserted-by":"crossref","unstructured":"K. Hiderah,\nThe truncated Euler\u2013Maruyama method of one-dimensional stochastic differential equations involving the local time at point zero,\nRandom Oper. Stoch. Equ. 31 (2023), no. 2, 141\u2013152.","DOI":"10.1515\/rose-2023-2003"},{"key":"2024022618254286523_j_mcma-2023-2018_ref_015","doi-asserted-by":"crossref","unstructured":"K. Hiderah and M. Bourza,\nCarath\u00e9odory approximate solutions for a class of perturbed reflected stochastic differential equations with irregular coefficients,\nStoch. Anal. Appl. 41 (2023), no. 3, 604\u2013625.","DOI":"10.1080\/07362994.2022.2064306"},{"key":"2024022618254286523_j_mcma-2023-2018_ref_016","doi-asserted-by":"crossref","unstructured":"D. Revuz and M. Yor,\nContinuous Martingales and Brownian Motion,\nGrundlehren Math. Wiss. 293,\nSpringer, Berlin, 1991.","DOI":"10.1007\/978-3-662-21726-9"},{"key":"2024022618254286523_j_mcma-2023-2018_ref_017","doi-asserted-by":"crossref","unstructured":"W. Yue and T. Zhang,\nAbsolute continuity of the laws of perturbed diffusion processes and perturbed reflected diffusion processes,\nJ. Theoret. 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