On the representation property for 1D general diffusion semimartingales

TL;DR

This paper characterizes when a one-dimensional diffusion semimartingale has the representation property, linking it to the absolute continuity of its scale function, with implications for market completeness and extremality in semimartingale problems.

Contribution

It establishes a necessary and sufficient condition for the representation property in diffusion semimartingales based on the scale function's regularity.

Findings

Representation property holds iff the scale function is locally absolutely continuous.

Laws of such diffusions are extreme points of their semimartingale problems.

Constructs a diffusion law that is not an extreme point.

Abstract

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 the laws of general diffusion semimartingales with such scale functions are extreme points of their semimartingale problems, and, moreover, we construct a general diffusion semimartingale whose law is no extreme point of its semimartingale problem. These observations contribute to a solution of problems posed by J. Jacod and M. Yor, and D. W. Stroock and M. Yor, on the extremality of strong Markov solutions.