{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:04:39Z","timestamp":1787321079370,"version":"build-2736575974"},"reference-count":25,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Control Optim."],"published-print":{"date-parts":[[2005,1]]},"abstract":"<jats:p>We study backward stochastic Riccati equations (BSREs) arising in quadratic optimal control problems with infinite dimensional stochastic differential state equations. We allow the coefficients, both in the state equation and in the cost, to be random. In such a context BSREs are backward stochastic differential equations existing in a non-Hilbert space and involving quadraticnonlinearities. We propose two different notions of solutions to BSREs and prove, for both of them, existence and uniqueness results. We also show that such solutions allow us to perform the synthesis of the optimal control. Finally we apply our results to the optimal control of a delay equation and of a wave equation with random damping.<\/jats:p>","DOI":"10.1137\/s0363012903425507","type":"journal-article","created":{"date-parts":[[2005,8,9]],"date-time":"2005-08-09T16:06:43Z","timestamp":1123603603000},"page":"159-194","source":"Crossref","is-referenced-by-count":40,"title":["On the Backward Stochastic Riccati Equation in Infinite Dimensions"],"prefix":"10.1137","volume":"44","author":[{"given":"Giuseppina","family":"Guatteri","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Gianmario","family":"Tessitore","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,26]]},"reference":[{"key":"R1","doi-asserted-by":"crossref","unstructured":"A. Bensoussan, G. da Prato, M. C. Delfour, and S. K. Mitter,\n                      Representation and Control of Infinite Dimensional Systems\n                      , Vol. 1, Syst. Control Found. Appl., Birkhuser, Boston, MA, 1992.","DOI":"10.1007\/978-1-4612-2750-2"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1137\/0314028"},{"key":"R3","doi-asserted-by":"crossref","unstructured":"J.\u2010M. Bismut,\n                      Contr\u00f4le des syst\u00e8mes lin\u00e9aires quadratiques: applications de l\u2019int\u00e9grale stochastique\n                      , in S\u00e9minaire de Probabilit\u00e9s, XII, Lecture Notes in Math. 649, Springer\u2010Varlag, Berlin, 1978.","DOI":"10.1007\/BFb0064606"},{"key":"R4","first-page":"634","volume":"26","author":"Chojnowska\u2010Michalik A.","year":"1978","journal-title":"Bull. Pol. Acad. Sci. Ser. Sci. Math."},{"key":"R5","doi-asserted-by":"crossref","unstructured":"G. da Prato and J. Zabczyk,\n                      Stochastic Equations in Infinite Dimensions\n                      , Cambridge University Press, Cambridge, UK, 1992.","DOI":"10.1017\/CBO9780511666223"},{"key":"R6","doi-asserted-by":"crossref","unstructured":"G. da Prato and J. Zabczyk,\n                      Ergodicity for Infinite\u2010Dimensional Systems\n                      , London Math. Soc. Lecture Note Ser. 229, Cambridge University Press, Cambridge, UK, 1996.","DOI":"10.1017\/CBO9780511662829"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1214\/aop\/1029867132"},{"key":"R8","unstructured":"M. Fuhrman and G. Tessitore,\n                      Generalized directional gradients, backward stochastic differential equations and mild solutions of semilinear parabolic equations\n                      , Appl. Math. Optim., to appear."},{"key":"R9","doi-asserted-by":"crossref","unstructured":"J. Hale,\n                      Theory of Functional Differential Equations\n                      , Appl. Math. Sci. 3, Springer\u2010Verlag, New York, 1977.","DOI":"10.1007\/978-1-4612-9892-2"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1080\/07362999108809250"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1214\/aop\/1019160253"},{"key":"R12","doi-asserted-by":"crossref","unstructured":"M. Kohlmann and S. Tang,\n                      New developments in backward stochastic Riccati equations and their applications\n                      , in Mathematical Finance (Konstanz, 2000), Trends Math., Birkhuser, Basel, 2001.","DOI":"10.1007\/978-3-0348-8291-0_19"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1016\/S0304-4149(01)00133-8"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1137\/S0363012900378760"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1137\/S036301299834973X"},{"key":"R16","doi-asserted-by":"crossref","unstructured":"A. Lunardi,\n                      Analytic Semigroups and Optimal Regularity in Parabolic Problems\n                      , Progr. Nonlinear Differential Equations Appl. 16, Birkh\u00e4user Verlag, Basel, 1995.","DOI":"10.1007\/978-3-0348-0557-5"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1080\/17442509808834149"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1016\/0167-6911(90)90082-6"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1137\/0330018"},{"key":"R20","doi-asserted-by":"crossref","unstructured":"S. Peng,\n                      Open problems on backward stochastic differential equations\n                      , in Control of Distributed Parameter and Stochastic Systems (Hangzhou, 1998), Kluwer, Boston, 1999.","DOI":"10.1007\/978-0-387-35359-3_32"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1137\/S0363012901387550"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1137\/0330040"},{"key":"R23","unstructured":"J. Yong and X. Zhou,\n                      Stochastic Controls: Hamiltonian Systems and HJB Equations\n                      , Springer\u2010Verlag, Berlin, 1999."},{"key":"R24","doi-asserted-by":"crossref","unstructured":"J. Zabczyk,\n                      Parabolic equations on Hilbert spaces\n                      , in Stochastic PDE\u2019s and Kolmogorov Equations in Infinite Dimensions, G. da Prato, ed., Lecture Notes in Math. 1715, Springer\u2010Verlag, Berlin, 1999, pp. 117\u2013213.","DOI":"10.1007\/BFb0092419"},{"key":"R25","doi-asserted-by":"publisher","DOI":"10.1007\/s002450010003"}],"container-title":["SIAM Journal on Control and Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/S0363012903425507","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:04:43Z","timestamp":1787317483000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/S0363012903425507"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2005,1]]},"references-count":25,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2005,1]]}},"alternative-id":["10.1137\/S0363012903425507"],"URL":"https:\/\/doi.org\/10.1137\/s0363012903425507","relation":{},"ISSN":["0363-0129","1095-7138"],"issn-type":[{"value":"0363-0129","type":"print"},{"value":"1095-7138","type":"electronic"}],"subject":[],"published":{"date-parts":[[2005,1]]}}}