Time homogeneous diffusions with a given marginal at a deterministic time

In this article, it is proved that for any probability law μ over R with finite first moment and a given deterministic time t > 0 , there exists a gap diffusion with law μ at the prescribed time t . thod starts by constructing a discrete time process X on a finite state space, where X τ has law μ , for a geometric time τ , independent of the diffusion. This argument is developed, using a fixed point theorem, to give conditions for the existence of a process with prescribed law when stopped at an independent time with negative binomial distribution. Reducing the time mesh gives a continuous time diffusion with prescribed law for τ with Gamma distribution. Keeping E [ τ ] = t fixed, the parameters of the Gamma distribution are altered, giving the prescribed law for the deterministic time. An approximating sequence establishes the result for arbitrary probability measure over R .