Using the definition from: Statistics of Random Processes 1 by Shiryaev, we have that an Ito process $\{X_t\}_{0\leq t\leq T}$, is called an Ito process of the diffusion type if we have:
$X_t = X_0 + \int_0^tA(s,x)ds + \int_0^tB(s,x)dW_s, \quad 0\leq t \leq T$
Where $A(t,x)$ and $B(t,x)$ are $\mathcal{F}_t^X$ measurable functionals.
Pretty much every reference I can find for diffusions covers in detail the Markovian case where the drift and diffusion coefficient depend on the current time and current state $(t, X_t)$.
My question is if there are any good references out there for the case where the coefficients depend on the and the full history $(t,X_{s\leq t})$. Specifically do we have results analogous to Kolmogorov forward-backward equations in the non-Markovian case?
Thanks.