On the non-equilibrium density of geometric mean reversion

The geometric mean reversion process X(⋅)X(⋅) is well known to play a fundamental role in economic dynamic models. While it is known, at least since Merton (1975), that the equilibrium distribution of geometric mean reversion, i.e. the distribution of X(∞)X(∞), is a gamma distribution, an explicit expression for the non-equilibrium distribution, i.e. the distribution of X(t)X(t) for t<∞t<∞, has not been known. The main result of this article is an analytic formula which computes the probability density function of X(t)X(t).