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We consider a controlled state equation of parabolic type on the halfline (0,+∞) with boundary conditions of Dirichlet type in which the unknown is equal to the sum of the control and of a white noise in time. We study finite horizon and infinite horizon optimal control problem related by means of backward stochastic differential equations.

A Stochastic Optimal Control Problem for the Heat Equation on the Halfline with Dirichlet Boundary-Noise and Boundary-Control