Semimartingale decomposition of convex functions of continuous semimartingales by brownian perturbation
ESAIM: Probability and Statistics, Tome 17 (2013), pp. 293-306.

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 ) d M , where H ( x ) is some particular measurable choice of subgradient ¯ 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 X ˜ = X + ϵ B , 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.

DOI : 10.1051/ps/2011146
Classification : 60H05
Mots clés : Itô's lemma, continuous semimartingales, convex functions
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Grinberg, Nastasiya F. Semimartingale decomposition of convex functions of continuous semimartingales by brownian perturbation. ESAIM: Probability and Statistics, Tome 17 (2013), pp. 293-306. doi : 10.1051/ps/2011146. http://www.numdam.org/articles/10.1051/ps/2011146/

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