Mathematics – Probability
Scientific paper
2008-02-04
Annals of Probability 2010, Vol. 38, No. 1, 76-101
Mathematics
Probability
Published in at http://dx.doi.org/10.1214/09-AOP476 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Scientific paper
10.1214/09-AOP476
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.
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