Time Homogeneous Diffusions with a Given Marginal at a Deterministic Time

In this article, it is proved that for any cumulative distribution function with compact support and a specified t > 0, there exists a diffusion martingale which has this law at time t. The article proves existence; no claims are made about uniqueness. After a discussion on strings and associated semigroups, the article gives a re-working of a standard approach to the problem of constructing an explicit discrete time martingale diffusion on a finite state space which, for a random geometrically distributed time that is independent of the diffusion, the law of the diffusion stopped at this random time has the prescribed law. This argument is developed, using a fixed point theorem, to determine conditions under which there is a discrete time martingale diffusion that has a prescribed law at an independent random time with negative binomial distribution. The step length for the time discretisation is then reduced and in the limit it is shown that for a finite state space, there exists a continuous time martingale diffusion such that $X_\tau$ has law $\mu$, where $\tau$ has a Gamma distribution. For fixed $t = E[\tau]$, the parameters of the Gamma distribution may be altered, reducing the coefficient of variation of $\tau$ to zero, to show that there is a martingale diffusion $X$ such that $X_t$ has law $\mu$. The argument is then extended to obtain the result for any state space that is a bounded measurable subset of $R$.