Can time-homogeneous diffusions produce any distribution? (Q1950378)

A scalar generalized diffusion is constructed by random time change of a scalar Brownian motion which depends on a prescribed non-negative `speed' measure. For a probability measure on line with finite mean, a speed measure can be chosen so that the corresponding generalized diffusion is a martingale whose time one distribution is the given probability measure. The proof is based on a construction for finitely supported speed measures, followed by a limiting argument for the general case. Application to an inverse problem in mathematical finance is given, followed by a characterization of speed measures which lead to generalized diffusions that are local martingales.