Feynman-kac Formulas for Black-scholes Type Operators

There are many references showing that a classical solution to the Black–Scholes equation is a stochastic solution. However, it is the converse of this theorem which is most relevant in applications and the converse is also more mathematically interesting. In the present article we establish such a converse. We find a Feynman–Kac type theorem showing that the stochastic representation yields a classical solution to the corresponding Black–Scholes equation with appropriate boundary conditions under very general conditions on the coefficients. We also obtain additional regularity results in the one-dimensional case.