We are differing in our initial assumptions: mine is that we cannot model transition densities of X alone.

Example 1:
dR = sigma R dW1
dsigma = nu dW2
dW1 dW2 = 0
The backward Kolmogorov equation is

dV/dt + 0.5*(sigma*R)^2*(d/dR)(d/dR)V + 0.5*nu^2*(d/dsigma)(d/dsigma)V = 0

V = payoff at t=T

Since V depends on sigma in a nonlinear fashion, the last term must enter the price, albeit at second order (for nu small). Re-stated, since I can (in principle) hedge my vega risk, the correct mathematical price is obtained assuming that I DO hedge this risk; but to hedge this risk I have a systematic gain/loss accoding to the "vol-gamma" term.

We are differing in our initial assumptions: mine is that we cannot model transition densities of X alone. >>
I agree. My assumption is that we can model transition densities directly without ever mentioning such a thing as volatility (something in the spirit of Rubinstein's tree but in the multiperiod setting).