The Annals of Probability

Nondifferentiable functions of one-dimensional semimartingales

George Lowther

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Abstract

We consider decompositions of processes of the form Y=f(t, Xt) 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.

Article information

Source
Ann. Probab. Volume 38, Number 1 (2010), 76-101.

Dates
First available in Project Euclid: 25 January 2010

Permanent link to this document
http://projecteuclid.org/euclid.aop/1264433993

Digital Object Identifier
doi:10.1214/09-AOP476

Zentralblatt MATH identifier
05678674

Mathematical Reviews number (MathSciNet)
MR2599194

Subjects
Primary: 60H05: Stochastic integrals 60G48: Generalizations of martingales 60G44: Martingales with continuous parameter 60G20: Generalized stochastic processes

Keywords
Semimartingale Itô’s lemma quadratic variation covariation Dirichlet process

Citation

Lowther, George. Nondifferentiable functions of one-dimensional semimartingales. Ann. Probab. 38 (2010), no. 1, 76--101. doi:10.1214/09-AOP476. http://projecteuclid.org/euclid.aop/1264433993.


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