  The goal of the paper is to show, under possibly weak assumptions, that the
function given by the Feynman-Kac formula is a classical solution of the
associated Kolmogorov equation. We also show that although this solution is
unbounded it fulfills Schauder's estimates. The paper collects mostly known
results on the regularity of solutions of linear parabolic equations of second
order and the regularity of solutions of Ito stochastic differential equations.
We aim to reproduce these and related results in a unified fashion.
