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Semimartingale decomposition of convex functions of continuous semimartingales by Brownian perturbation

Published online by Cambridge University Press:  17 May 2013

Nastasiya F. Grinberg
Nastasiya F. Grinberg*
Affiliation:
Department of Statistics, University of Warwick, Coventry, CV4 7AL, UK. N.F.Grinberg@gmail.com
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Abstract

In this note we prove that the local martingale part of a convex function f of a d-dimensional semimartingale X = M + A can be written in terms of an Itô stochastic integral ∫H(X)dM, where H(x) is some particular measurable choice of subgradient \hbox{$\sub$}∇f(x) of f at x, and M is the martingale part of X. This result was first proved by Bouleau in [N. Bouleau, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981) 87–90]. Here we present a new treatment of the problem. We first prove the result for \hbox{$\widetilde{X}=X+\epsilon B$}􏽥X=X+ϵB, ϵ > 0, where B is a standard Brownian motion, and then pass to the limit as ϵ → 0, using results in [M.T. Barlow and P. Protter, On convergence of semimartingales. In Séminaire de Probabilités, XXIV, 1988/89, Lect. Notes Math., vol. 1426. Springer, Berlin (1990) 188–193; E. Carlen and P. Protter, Illinois J. Math. 36 (1992) 420–427]. The former paper concerns convergence of semimartingale decompositions of semimartingales, while the latter studies a special case of converging convex functions of semimartingales.

Keywords

Itô’s lemmacontinuous semimartingalesconvex functions.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 17 , 2013 , pp. 293 - 306
Copyright
© EDP Sciences, SMAI, 2013

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References

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