BSDEs with Singular Terminal Condition and a Control Problem with Constraints

We provide a probabilistic solution of a not necessarily Markovian control problem with a state constraint by means of a backward stochastic differential equation (BSDE). The novelty of our solution approach is that the BSDE possesses a singular terminal condition. We prove that a solution of the BSDE exists, thus partly generalizing existence results obtained by Popier in [Stochastic Process. Appl., 116 (2006), pp. 2014--2056] and [Ann. Probab., 35 (2007), pp. 1071--1117]. We perform a verification and discuss special cases for which the control problem has explicit solutions.