Nondifferentiable Functions of One Dimensional Semimartingales

In this paper we consider decompositions of processes of the form Y = f(t,Xt) where X is a one dimensional semimartingale, but f is not required to be differentiable so Ito’s formula does not apply. First, in the case where f(t, x) is independent of t, we show that requiring it to be locally Lipschitz continuous in x is enough for an Ito style decomposition to apply. This decomposes Y into a stochastic integral term and a term whose quadratic variation is well defined and has zero continuous part. For the time dependent case we show that the same decomposition still holds under the additional conditions that the left and right derivatives of f(t, x) in x exist, it is right-continuous in t, and that locally its variation with respect to t is integrable in x. In particular, in the continuous case this shows that Y is a Dirichlet process. We furthermore prove that such processes satisfy a decomposition into continuous martingale and purely discontinuous terms, and a Doob-Meyer style decomposition.