Partial Schauder estimates for unbounded solutions of the Kolmogorov equation

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.