On Classical Solutions of Linear Stochastic Integro-Differential   Equations
  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. 
James-Michael Leahy, Remigijus Mikulevicius
math.PR math.AP