On the representation property for 1D general diffusion semimartingales

A general diffusion semimartingale is a one-dimensional path-continuous semimartingale that is also a regular strong Markov process. We say that a continuous semimartingale has the representation property if all local martingales w.r.t. its canonical filtration have an integral representation w.r.t. its continuous local martingle part. The representation property is of fundamental interest in the field of mathematical finance, where it is strongly connected to market completeness. The main result from this paper shows that the representation property holds for a general diffusion semimartingale (that is not started in an absorbing boundary point) if and only if its scale function is (locally) absolutely continuous on the interior of the state space. As an application of our main theorem, we deduce that general diffusion semimartingales with such scale functions are extreme points in their semimartingale problem. This observation contributes to a solution of a problem posed by J. Jacod and M. Yor on the extremality of strong Markov solutions.