Multiplicative Langevin Process for Volatilities Produces Observed Q-Variance Regularities

Q-variance (so-called) posits a statistical relationship $\mathbf{E}(σ^2 | z) = σ_0^2 + \tfrac{1}{2}z^2$ between an asset's volatility $σ^2$, as observed in a time interval $T$, and its (suitably scaled) return $z$ in the same interval. We here show that this relationship is {\em exactly equivalent} to to positing an Inverse Gamma probability distribution for $σ^2$ itself. We then show that such a distribution is exactly generated by a multiplicative Langevin process with an arbitrary, settable coherence time $τ_c$, so that very nearly the same Q-variance relationship will hold for all $T \ll τ_c$.