{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:33:11Z","timestamp":1787319191585,"version":"build-2736575974"},"reference-count":30,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"7","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Control Optim."],"published-print":{"date-parts":[[2010,1]]},"abstract":"<jats:p>We consider an It\u00f4 stochastic differential equation with delay, driven by Brownian motion, whose solution, by an appropriate reformulation, defines a Markov process X with values in a space of continuous functions $\\mathbf{C}$, with generator $\\mathcal{L}$. We then consider a backward stochastic differential equation depending on X, with unknown processes $(Y,Z)$, and we study properties of the resulting system, in particular we identify the process Z as a deterministic functional of X. We next prove that the forward-backward system provides a suitable solution to a class of parabolic partial differential equations on the space $\\mathbf{C}$ driven by $\\mathcal{L}$, and we apply this result to prove a characterization of the fair price and the hedging strategy for a financial market with memory effects. We also include applications to optimal stochastic control of differential equation with delay: in particular we characterize optimal controls as feedback laws in terms of the process X.<\/jats:p>","DOI":"10.1137\/080730354","type":"journal-article","created":{"date-parts":[[2010,8,17]],"date-time":"2010-08-17T18:39:40Z","timestamp":1282070380000},"page":"4624-4651","source":"Crossref","is-referenced-by-count":43,"title":["Stochastic Equations with Delay: Optimal Control via BSDEs and Regular Solutions of Hamilton\u2013Jacobi\u2013Bellman Equations"],"prefix":"10.1137","volume":"48","author":[{"given":"Marco","family":"Fuhrman","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Federica","family":"Masiero","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Gianmario","family":"Tessitore","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2010,8,17]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1080\/07362990601139669"},{"key":"R2","unstructured":"J. P. Aubin and H. Frankowska,\n                      Set-Valued Analysis\n                      , Birkh\u00e4user Boston, Boston, 1990."},{"key":"R3","doi-asserted-by":"crossref","unstructured":"T. Bj\u00f6rk,\n                      Arbitrage Theory in Continuous Time\n                      , Oxford University Press, Oxford, 1998.","DOI":"10.1093\/0198775180.001.0001"},{"key":"R4","doi-asserted-by":"crossref","unstructured":"M.H. Chang,\n                      Stochastic Control of Hereditary Systems and Applications\n                      , Stoch. Mod. Appl. Probab. 59, Springer, New York, 2008.","DOI":"10.1007\/978-0-387-75816-9"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1007\/s00245-007-9003-z"},{"key":"R6","doi-asserted-by":"crossref","unstructured":"G. Da Prato and J. Zabczyk,\n                      Stochastic Equations in Infinite Dimensions\n                      , Encyclopedia Math. Appl. 44, Cambridge University Press, Cambridge, UK, 1992.","DOI":"10.1017\/CBO9780511666223"},{"key":"R7","doi-asserted-by":"crossref","unstructured":"G. Da Prato and J. Zabczyk,\n                      Ergodicity for Infinite-Dimensional Systems\n                      , London Math. Soc. Lecture Note Ser. 229, Cambridge University Press, Cambridge, UK, 1996.","DOI":"10.1017\/CBO9780511662829"},{"key":"R8","doi-asserted-by":"crossref","unstructured":"N. El Karoui and L. Mazliak, eds.\n                      Backward Stochastic Differential Equations\n                      , Pitman Res. Notes Math. Ser. 364, Longman, Harlow, UK, 1997.","DOI":"10.1007\/BFb0092001"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1111\/1467-9965.00022"},{"key":"R10","unstructured":"W. H. Fleming and H. M. Soner,\n                      Controlled Markov Processes and Viscosity Solutions\n                      , Appl. Math. (N.Y.) 25, Springer-Verlag, New York, 1993."},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1214\/aop\/1029867132"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1214\/aop\/1079021459"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1007\/s00245-004-0814-x"},{"key":"R14","doi-asserted-by":"crossref","unstructured":"J. Hale,\n                      Theory of Functional Differential Equations\n                      , Appl. Math. Sci. 3, Springer-Verlag, New York, 1971.","DOI":"10.1007\/978-1-4615-9968-5"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1080\/07362999308809301"},{"key":"R16","doi-asserted-by":"crossref","unstructured":"S. Kusuoka and D. Stroock,\n                      Applications of the Malliavin calculus.\n                      I, in Stochastic Analysis (Katata\/Kyoto, 1982), North\u2013Holland Math. Library 32, North\u2013Holland, Amsterdam, 1984, pp. 271\u2013306.","DOI":"10.1016\/S0924-6509(08)70397-0"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1137\/050632725"},{"key":"R18","unstructured":"S. E. A. Mohammed,\n                      Stochastic Functional Differential Equations\n                      , Res. Notes Math. 99, Pitman, Boston, 1984."},{"key":"R19","doi-asserted-by":"crossref","unstructured":"S. E. A. Mohammed,\n                      Stochastic Differential Systems with Memory: Theory, Examples and Applications\n                      , in Stochastic Analysis and Related Topics. VI. Proceedings of the 6th Oslo-Silivri Workshop (Geilo, 1996), Progr. Probab. 42, Birkh\u00e4user Boston, Boston, 1998.","DOI":"10.1007\/978-1-4612-2022-0_1"},{"key":"R20","doi-asserted-by":"crossref","unstructured":"M. Musiela and M. Rutkowski,\n                      Martingale Methods in Financial Modelling\n                      , Appl. Math. (N.Y.) 36, Springer Verlag, Berlin, 1997.","DOI":"10.1007\/978-3-662-22132-7"},{"key":"R21","doi-asserted-by":"crossref","unstructured":"D. Nualart,\n                      The Malliavin Calculus and Related Topics\n                      , Probab. Appl., Springer-Verlag (N.Y.), New York, 1995.","DOI":"10.1007\/978-1-4757-2437-0"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1007\/BF00353876"},{"key":"R23","doi-asserted-by":"crossref","unstructured":"\u00c9. Pardoux,\n                      BSDE's, weak convergence and homogeneization of semilinear PDE's\n                      , in Nonlinear Analysis, Differential Equations and Control, F. H. Clarke and R. J. Stern, eds., Kluwer Academic Publishers, Dordrecht, 1999, pp. 503\u2013549.","DOI":"10.1007\/978-94-011-4560-2_9"},{"key":"R24","doi-asserted-by":"publisher","DOI":"10.1016\/0167-6911(90)90082-6"},{"key":"R25","doi-asserted-by":"crossref","unstructured":"\u00c9. Pardoux and S. Peng,\n                      Backward stochastic differential equations and quasilinear parabolic partial differential equations\n                      , in Stochastic Partial Differential Equations and Their Applications, Lecture Notes in Control and Inform. Sci. 176, B. L. Rozowskii and R. B. Sowers, eds., Springer, Berlin, 1992, pp. 200\u2013217.","DOI":"10.1007\/BFb0007334"},{"key":"R26","doi-asserted-by":"publisher","DOI":"10.1007\/BF01195073"},{"key":"R27","doi-asserted-by":"publisher","DOI":"10.1016\/0304-4149(95)93237-A"},{"key":"R28","doi-asserted-by":"publisher","DOI":"10.1007\/BF01303801"},{"key":"R29","doi-asserted-by":"publisher","DOI":"10.1080\/17442500008834244"},{"key":"R30","unstructured":"P. Willmott, J. Dewynne, and S. Howison,\n                      Option Pricing\n                      , Oxford Financial Press, Oxford, 1993."}],"container-title":["SIAM Journal on Control and Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/080730354","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T12:46:13Z","timestamp":1787316373000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/080730354"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,1]]},"references-count":30,"journal-issue":{"issue":"7","published-print":{"date-parts":[[2010,1]]}},"alternative-id":["10.1137\/080730354"],"URL":"https:\/\/doi.org\/10.1137\/080730354","relation":{},"ISSN":["0363-0129","1095-7138"],"issn-type":[{"value":"0363-0129","type":"print"},{"value":"1095-7138","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,1]]}}}