Suppose that $X_t$ is a stochastic process described by linear SDE
$$dX_t = \big( a(t) X_t + b(t) \big) dt + \big( c(t) X_t + d(t) \big) dB_t$$
For some function $g(x)$ define new function $f(x) := \mathbb{E}(g(X_t)|X_s = x), t > s$
I want to know under what regularity conditions on functions $a,b,c,d, g$ function $f$ is differentiable/weakly differentiable?
I tried to google some information about this problem, but unsuccessfully