Semimartingale decomposition of convex functions of continuous semimartingales by brownian perturbation

Grinberg, Nastasiya F.

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 $\int H\left(X\right)dM$, where $H\left(x\right)$ is some particular measurable choice of subgradient $\overline{\nabla }f\left(x\right)$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 $\tilde{X}=X+ϵB,ϵ>0$, where $B$ is a standard Brownian motion, and then pass to the limit as $ϵ\to 0$, 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.

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