We investigate the linear programming framework for an exit-time stochastic control problem and apply the moment-sum-of-squares hierarchy to obtain tight pointwise bounds and global bounding functions for the value function. The primal linear program over suitable measures and the dual linear program over test functions are implemented numerically by semidefinite programs which target at moments and sum-of-squares polynomial representations, respectively. Numerically optimized bounds converge to the value function from below as polynomial degree increases to infinity under suitable technical conditions. We focus on the dual problem, which is particularly effective, as its single implementation yields a polynomial bounding function over the entire problem domain, and since it allows a flexible choice of objective function, one may improve the global bound on regions of interest.