  We solve explicitly the following problem: for a given probability measure
mu, we specify a generalised martingale diffusion X which, stopped at an
independent exponential time T, is distributed according to mu. The process X
is specified via its speed measure m. We present three proofs. First we show
how the result can be derived from the solution of Bertoin and Le Jan (1992) to
the Skorokhod embedding problem. Secondly, we give a proof exploiting
applications of Krein's spectral theory of strings to the study of linear
diffusions. Finally, we present a novel direct probabilistic proof based on a
coupling argument.
