Abstract
We consider functions,F, of a semimartingale,X, on a complete manifold which fail to beC2 only on, and are sufficiently well-behaved near, a codimension 1 subset ℒ. We obtain an extension of the Ito formula which is valid for all time by adding a continuous predictable process given explicitly in terms of two geometric local times ofX on ℒ and the Gâteaux derivative ofF. 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 semimartingale.
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