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{$\sub$} 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{$\widetilde{X}=X+\epsilon B$},
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.