Statistics and Analytical Sciences
It is proven, under a set of assumptions differing from the usual ones in the unboundedness of the time interval, that, in an economy in equilibrium consisting of a risk-free cash account and an equity whose price process is a geometric Brownian motion on [0,∞), the drift rate must be close to the risk-free rate; if the drift rate μ and the risk-free rate r are constants, then r = μ and the price process is the same under both empirical and risk neutral measures. Contributing in some degree perhaps to interest in this mathematical curiosity is the fact, based on empirical data taken at various times over an assortment of equities and relatively short durations, that no tests of the hypothesis of equality are rejected.
Gadidov, Anda, and M. C. Spruill. 2011. "Drift and the Risk-Free Rate." Journal Of Probability & Statistics 1-19.