  In this paper we derive semi-closed form prices of barrier (perhaps,
time-dependent) options for the Hull-White model, ie., where the underlying
follows a time-dependent OU process with a mean-reverting drift. Our approach
is similar to that in (Carr and Itkin, 2020) where the method of generalized
integral transform is applied to pricing barrier options in the time-dependent
OU model, but extends it to an infinite domain (which is an unsolved problem
yet). Alternatively, we use the method of heat potentials for solving the same
problems. By semi-closed solution we mean that first, we need to solve
numerically a linear Volterra equation of the first kind, and then the option
price is represented as a one-dimensional integral. Our analysis shows that
computationally our method is more efficient than the backward and even forward
finite difference methods (if one uses them to solve those problems), while
providing better accuracy and stability.
