  In this paper we develop the formalism for the stochastic approach to
inflation at all order in slow-roll parameters. This is done by including the
momentum and Hamiltonian constraints into the stochastic equations. We then
specialise to the widely used Starobinski approximation where interactions
between IR and UV modes are neglected. We show that, whenever this
approximation holds, no significant deviations are observed when comparing the
two-point correlation functions (power spectrum) calculated with stochastic
methods, to the ones calculated with the QFT approach to linear theory. As a
byproduct, we argue that: a) the approaches based on the Starobinski
approximation, generically, do not capture any loop effects of the quantum
scalar-gravity system; b) correlations functions can only be calculated in the
linear theory regimes, thus, no non-perturbative statistics can be extracted
within this approximation, as commonly claimed.
