{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,4]],"date-time":"2026-06-04T09:32:02Z","timestamp":1780565522380,"version":"3.54.1"},"reference-count":22,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2020,11,26]],"date-time":"2020-11-26T00:00:00Z","timestamp":1606348800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Shandong  Provincial Natural Science Foundation","award":["ZR2019ZD42"],"award-info":[{"award-number":["ZR2019ZD42"]}]},{"name":"the Natural Science Foundation of China","award":["11831010, 61961160732"],"award-info":[{"award-number":["11831010, 61961160732"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper we study the existence and uniqueness of solutions for one kind of backward doubly stochastic differential equations (BDSDEs) with Markov chains. By generalizing the It\u00f4\u2019s formula, we study such problem under the Lipschitz condition. Moreover, thanks to the Yosida approximation, we solve such problem under monotone condition. Finally, we give the comparison theorems for such equations under the above two conditions respectively.<\/jats:p>","DOI":"10.3390\/sym12121953","type":"journal-article","created":{"date-parts":[[2020,11,26]],"date-time":"2020-11-26T22:00:33Z","timestamp":1606428033000},"page":"1953","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Backward Doubly Stochastic Differential Equations with Markov Chains and a Comparison Theorem"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8403-9812","authenticated-orcid":false,"given":"Ning","family":"Ma","sequence":"first","affiliation":[{"name":"School of Mathematics, Shandong University, Jinan 250100, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0758-9463","authenticated-orcid":false,"given":"Zhen","family":"Wu","sequence":"additional","affiliation":[{"name":"School of Mathematics, Shandong University, Jinan 250100, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2020,11,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1016\/0167-6911(90)90082-6","article-title":"Adapted solution of a backward stochastic differential equation","volume":"14","author":"Shige","year":"1990","journal-title":"Syst. Control. Lett."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1007\/BF01192514","article-title":"Backward doubly stochastic differential equations and systems of quasilinear SPDEs","volume":"98","author":"Shige","year":"1994","journal-title":"Probab. Theory Relat. Fields"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"425","DOI":"10.1016\/S0167-7152(96)00103-4","article-title":"Backward stochastic differential equations with continuous coefficient","volume":"32","author":"Lepeltier","year":"1997","journal-title":"Stat. Probab. Lett."},{"key":"ref_4","first-page":"481","article-title":"Backward stochastic differential equations with locally Lipschitz coefficient","volume":"333","author":"Khaled","year":"2001","journal-title":"C. R. l\u2019Acad\u00e9mie Sci. Ser. Math."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"939","DOI":"10.1081\/SAP-120037626","article-title":"Reflected backward stochastic differential equation with locally monotone coefficient","volume":"22","author":"Bahlali","year":"2004","journal-title":"Stoch. Anal. Appl."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"759","DOI":"10.1016\/j.jde.2011.05.017","article-title":"BDSDEs with locally monotone coefficients and Sobolev solutions for SPDEs","volume":"251","author":"Wu","year":"2011","journal-title":"J. Differ. Equ."},{"key":"ref_7","first-page":"227","article-title":"Existence for BSDE with superlinear\u2013quadratic coefficient","volume":"63","author":"Lepeltier","year":"1998","journal-title":"Stoch. Int. J. Probab. Stoch. Process."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"558","DOI":"10.1214\/aop\/1019160253","article-title":"Backward stochastic differential equations and partial differential equations with quadratic growth","volume":"28","author":"Kobylanski","year":"2000","journal-title":"Ann. Probab."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"953","DOI":"10.1016\/j.jde.2009.12.013","article-title":"Stationary solutions of SPDEs and infinite horizon BDSDEs with non-Lipschitz coefficients","volume":"248","author":"Zhang","year":"2010","journal-title":"J. Differ. Equ."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1111\/1467-9965.00022","article-title":"Backward Stochastic Differential Equations in Finance","volume":"7","author":"Peng","year":"1997","journal-title":"Math. Financ."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1214\/ECP.v5-1025","article-title":"A Converse Comparison Theorem for BSDEs and Related Properties of g-Expectation","volume":"5","author":"Briand","year":"2000","journal-title":"Electron. Commun. Probab."},{"key":"ref_12","first-page":"135","article-title":"On the comparison theorem for multidimensional BSDEs","volume":"343","author":"Hu","year":"2006","journal-title":"Probab. Theory"},{"key":"ref_13","first-page":"674","article-title":"The Comparison Theorem of Forward-Backward Stochastic Differential Equations with Jumps and with Random Terminal Time","volume":"10","author":"Yin","year":"2002","journal-title":"Math. Prepr. Arch."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1081\/SAP-200044444","article-title":"Comparison theorems of backward doubly stochastic differential equations and applications","volume":"23","author":"Shi","year":"2005","journal-title":"Stoch. Anal. Appl."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"426","DOI":"10.1016\/j.spl.2008.09.011","article-title":"Comparison theorems for forward backward SDEs","volume":"79","author":"Wu","year":"2009","journal-title":"Stat. Probab. Lett."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1016\/j.jmaa.2013.05.011","article-title":"BSDEs with regime switching: Weak convergence and applications","volume":"407","author":"Tao","year":"2013","journal-title":"J. Math. Anal. Appl."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1137\/130938967","article-title":"Optimal switching under a regime-switching model with two-time-scale markov chains","volume":"13","author":"Tao","year":"2009","journal-title":"Multiscale Model. Simul."},{"key":"ref_18","first-page":"2041","article-title":"Reflected and doubly reflected BSDEs with jumps: A priori estimates and comparison","volume":"18","author":"Matoussi","year":"2008","journal-title":"Ann. Appl. Probab."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Cr\u00e9pey, S. (2011). About the Pricing Equations in Finance. Paris-Princeton Lectures on Mathematical Finance 2010, Springer.","DOI":"10.1007\/978-3-642-14660-2_2"},{"key":"ref_20","unstructured":"Ikeda, N., and Watanabe, S. (1989). Stochastic Differential Equations and Diffusion Processes, Elsevier."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"535","DOI":"10.1007\/BF00353876","article-title":"Stochastic calculus with anticipating integrands","volume":"78","author":"Nualart","year":"1988","journal-title":"Probab. Theory Relat. Fields"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Chung, K., and Walsh, J.B. (2006). Markov Processes, Brownian Motion, and Time Symmetry, Springer.","DOI":"10.1007\/0-387-28696-9"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/12\/1953\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:37:35Z","timestamp":1760179055000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/12\/1953"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,11,26]]},"references-count":22,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2020,12]]}},"alternative-id":["sym12121953"],"URL":"https:\/\/doi.org\/10.3390\/sym12121953","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,11,26]]}}}