We study a linear quadratic problem for a system governed by the heat equation on a halfline with boundary control and Dirichlet boundary noise. We show that this problem can be reformulated as a stochastic evolution equation in a certain weighted $L^2$ space. An appropriate choice of weight allows us to prove a stronger regularity for the boundary terms appearing in the infinite dimensional state equation. The direct solution of the Riccati equation related to the associated nonstochastic problem is used to find the solution of the problem in feedback form and to write the value function of the problem.