{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:17:32Z","timestamp":1760059052902,"version":"build-2065373602"},"reference-count":15,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2025,5,16]],"date-time":"2025-05-16T00:00:00Z","timestamp":1747353600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100002383","name":"KSU Researchers Supporting Project","doi-asserted-by":"publisher","award":["RSPD2025R1075"],"award-info":[{"award-number":["RSPD2025R1075"]}],"id":[{"id":"10.13039\/501100002383","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>We focus on solving stochastic differential equations driven by jump processes (SDEJs) with measurable drifts that may exhibit quadratic growth. Our approach leverages a space transformation and It\u00f4-Krylov\u2019s formula to effectively eliminate the singular component of the drift, allowing us to obtain a transformed SDEJ that satisfies classical solvability conditions. By applying the inverse transformation proven to be a one-to-one mapping, we retrieve the solution to the original equation. This methodology offers several key advantages. First, it extends the well-known result of Le Gall (1984) from Brownian-driven SDEs to the jump process setting, broadening the range of applicable stochastic models. Second, it provides a robust framework for handling singular drifts, enabling the resolution of equations that would otherwise be intractable. Third, the approach accommodates drifts with quadratic growth, making it particularly relevant for financial modeling, insurance risk assessment, and other applications where such growth behavior is common. Finally, the inclusion of multiple examples illustrates the practical effectiveness of our method, demonstrating its flexibility and applicability to real-world problems.<\/jats:p>","DOI":"10.3390\/axioms14050374","type":"journal-article","created":{"date-parts":[[2025,5,16]],"date-time":"2025-05-16T05:11:57Z","timestamp":1747372317000},"page":"374","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Existence and Uniqueness of Solutions to SDEs with Jumps and Irregular Drifts"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0889-3387","authenticated-orcid":false,"given":"Mhamed","family":"Eddahbi","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Sciences, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2025,5,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"309","DOI":"10.1214\/aop\/1176994472","article-title":"On skew Brownian motion","volume":"9","author":"Harrison","year":"1981","journal-title":"Ann. Probab."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Portenko, N.I. (1990). Translations of Mathematical Monographs. Generalized Diffusion Processes, American Mathematical Society.","DOI":"10.1090\/mmono\/083"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Stroock, D.W., and Yor, M. (1981). Some remarkable martingales. S\u00e9minaire de Probabilit\u00e9s XV 1979\/80, Springer.","DOI":"10.1007\/BFb0088396"},{"key":"ref_4","first-page":"32","article-title":"Strong existence, uniqueness and non-uniqueness in an equation involving local time","volume":"17","author":"Barlow","year":"1983","journal-title":"S\u00e9minaire Probab. 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Stochastic Differential Systems Filtering and Control, Springer.","DOI":"10.1007\/BF00532642"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1002\/mana.19891430115","article-title":"Strong Markov Continuous Local Martingales and Solutions of One-Dimensional Stochastic Differential Equations (Part I)","volume":"143","author":"Engelbert","year":"1989","journal-title":"Math. Nachrichten"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1002\/mana.19911510111","article-title":"Strong Markov Continuous Local Martingales and Solutions of One-Dimensional Stochastic Differential Equations (Part III)","volume":"151","author":"Engelbert","year":"1991","journal-title":"Math. Nachrichten"},{"key":"ref_10","first-page":"19","article-title":"One-dimensional stochastic differential equations with singular and degenerate coefficients","volume":"67","author":"Bass","year":"2005","journal-title":"Sankhy\u0101 Indian J. 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Numerical solutions of stochastic differential equations with jumps and measurable drifts. Mathematics, 11.","DOI":"10.3390\/math11173755"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"5263","DOI":"10.2298\/FIL2215263E","article-title":"Numerical solution of quadratic SDE with measurable drift","volume":"36","author":"Eddahbi","year":"2022","journal-title":"Filomat"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/5\/374\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:33:36Z","timestamp":1760031216000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/5\/374"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,5,16]]},"references-count":15,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2025,5]]}},"alternative-id":["axioms14050374"],"URL":"https:\/\/doi.org\/10.3390\/axioms14050374","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,5,16]]}}}