Nondifferentiable functions of one-dimensional semimartingales

TL;DR

This paper extends Itô's calculus to nondifferentiable functions of one-dimensional semimartingales, establishing conditions under which such functions admit a decomposition into Dirichlet processes, with implications for diffusion theory.

Contribution

It introduces a generalized decomposition for nondifferentiable functions of semimartingales, broadening the scope of stochastic calculus beyond classical differentiability requirements.

Findings

Nondifferentiable functions can be decomposed into Dirichlet processes under Lipschitz conditions.

The decomposition applies even when the function depends on time and is discontinuous.

Results have potential applications in the theory of one-dimensional diffusions.

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 It\^{o}'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 It\^{o}-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.