ITO CORRECTION TERMS FOR THE RADIAL PARTS OF SEMIMARTINGALES ON MANIFOLDS

We consider functions, F, of a semimartingale, X, on a complete manifo ld which fail to be C-2 only on, and are sufficiently well-behaved nea r, a codimension 1 subset L. We obtain an extension of the Ito formula which is valid for all time by adding a continuous predictable proces s given explicitly in terms of two geometric local times of X on L and the Gateaux derivative of F. We then examine the cut locus of a point of the manifold in sufficient detail to show that this result applies to give a corresponding expression for the radial part of the semimar tingale.