{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:26:57Z","timestamp":1787318817212,"version":"build-2736575974"},"reference-count":23,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"5","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Control Optim."],"published-print":{"date-parts":[[2010,1]]},"abstract":"<jats:p>We study a class of Markovian optimal stochastic control problems in which the controlled process $Z^{\\nu}$ is constrained to satisfy an almost sure constraint $Z^{\\nu}(T)\\in G\\subset\\mathbb{R}^{d+1}$ $\\mathbb{P}$-a.s. at some final time $T&gt;0$. When the set is of the form $G:=\\{(x,y)\\in\\mathbb{R}^d\\times\\mathbb{R}:g(x,y)\\geq0\\}$, with g nondecreasing in y, we provide a Hamilton\u2013Jacobi\u2013Bellman characterization of the associated value function. It gives rise to a state constraint problem, where the constraint can be expressed in terms of an auxiliary value function w which characterizes the set $D:=\\{(t,Z^{\\nu}(t))\\in[0,T]\\times\\mathbb{R}^{d+1}:Z^{\\nu}(T)\\in G$ a.s. for some $\\nu\\}$. Contrary to standard state constraint problems, the domain D is not given a priori and we do not need to impose conditions on its boundary. It is naturally incorporated in the auxiliary value function w, which itself is a viscosity solution of a nonlinear parabolic PDE. Applying ideas recently developed in Bouchard, Elie, and Touzi [SIAM J. Control Optim., 48 (2009), pp. 3123\u20133150], our general result also allows us to consider optimal control problems with moment constraints of the form $\\mathbb{E}[g(Z^{\\nu}(T))]\\geq0$ or $\\mathbb{P}[g(Z^{\\nu}(T))\\geq0]\\geq p$.<\/jats:p>","DOI":"10.1137\/090757629","type":"journal-article","created":{"date-parts":[[2010,2,17]],"date-time":"2010-02-17T18:07:17Z","timestamp":1266430037000},"page":"3501-3531","source":"Crossref","is-referenced-by-count":52,"title":["Optimal Control under Stochastic Target Constraints"],"prefix":"10.1137","volume":"48","author":[{"given":"Bruno","family":"Bouchard","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Romuald","family":"Elie","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Cyril","family":"Imbert","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2010,2,17]]},"reference":[{"key":"R1","unstructured":"J.P. Aubin,\n                      Viability Theory\n                      , Systems Control Found. Appl., Birkh\u00e4user Boston, Boston, MA, 1991."},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1080\/03605309508821090"},{"key":"R3","unstructured":"D. Bertsekas and S. E. Shreve,\n                      Stochastic Optimal Control: The Discrete Time Case\n                      , Math. Sci. Engrg. 139, Academic Press, New York, London, 1978."},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1016\/S0304-4149(02)00129-1"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1137\/08073593X"},{"key":"R6","unstructured":"B. Bouchard and V. T. Nam,\n                      The American version of the geometric dynamic programming principle: Application to the pricing of American options under constraints\n                      , Appl. Math. 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Shreve,\n                      Methods of Mathematical Finance\n                      , Springer-Verlag, New York, 1998.","DOI":"10.1007\/b98840"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1016\/j.jedc.2003.11.005"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1512\/iumj.1994.43.43020"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1007\/BF01442856"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1214\/ECP.v12-1261"},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1137\/0324032"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1007\/s100970100039"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1137\/S0363012900378863"},{"key":"R23","unstructured":"J. Yong and X. Y. Zhou,\n                      Stochastic Controls, Hamiltonian Systems and HJB Equations\n                      , Appl. Math. (N. 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