Partial Schauder estimates for unbounded solutions of the Kolmogorov equation

Abstract It is well known that a classical solution of the Kolmogorov equation is given by the Feynman-Kac formula applied to a solution of the corresponding stochastic differential equation. The goal of the paper is to present, under possibly weak assumptions, a solution to the inverse problem that the Feynman-Kac formula, applied to a solution of a stochastic equation, defines a classical solution of the associated Kolmogorov equation. To solve this problem, we rely on many known results on the regularity of solutions of linear parabolic equations of second order and the regularity of solutions of Ito stochastic differential equations reformulated in a unified fashion. We supplement the existing results with new ones obtained under assumptions weaker than in the existing literature. We also show that although for linearly growing coefficients of the stochastic equation a solution of the Kolmogorov equation is unbounded with polynomial growth, it fulfills partial Schauder’s estimates.