  We prove the existence of classical solutions to parabolic linear stochastic
integro-differential equations with adapted coefficients using Feynman-Kac
transformations, conditioning, and the interlacing of space-inverses of
stochastic flows associated with the equations. The equations are forward and
the derivation of existence does not use the "general theory" of SPDEs.
Uniqueness is proved in the class of classical solutions with polynomial
growth.
