Nondifferentiable functions of one-dimensional semimartingales

Abstract

We consider decompositions of processes of the form Y=f(t, X_t) where X is a semimartingale. The function f is not required to be differentiable, so Itô’s lemma does not apply.

In the case where f(t, x) is independent of t, it is shown that requiring f to be locally Lipschitz continuous in x is enough for an Itô-style decomposition to exist. In particular, Y will be a Dirichlet process. We also look at the case where f(t, x) can depend on t, possibly discontinuously. It is shown, under some additional mild constraints on f, that the same decomposition still holds. Both these results follow as special cases of a more general decomposition which we prove, and which applies to nondifferentiable functions of Dirichlet processes.

Possible applications of these results to the theory of one-dimensional diffusions are briefly discussed.