Semimartingale decomposition of convex functions of continuous semimartingales by Brownian perturbation

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 Ito stochastic integral ∫ H ( X )d M , where H ( x ) is some particular measurable choice of subgradient 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 Ser. I Math. 292 (1981) 87–90]. Here we present a new treatment of the problem. We first prove the result for , 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 Seminaire de Probabilites, 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.