the distribution of D(i) must satisfy E(exp(D(i)) = Q, not 1; and one can achieve that even with a zero-mean Gaussian if desired. In that case one would have a legitimate log-Browninan model (with Gaussian increments) such that that E(Z(i0+n)) = Z(i0), E(P(i0+n)) = P(i0)*Q^n, and Prob(P(i0+n) < P(i0)) = 1/2. Does this make sense?
That's what all my models do, except I never use gaussians unless I need consistency with closed form options. Can't tell you if it's standard.
Regarding EMH, for every Rijksbank prize winner who advocates some formulation, I can name 2 who will repudiate that formulation as obviously inconsistent with basic observations. Use it every day, in order to do principled valuations, but give it no credence.