Semimartingale decomposition of convex functions of continuous semimartingales by brownian perturbation
In this note we prove that the local martingale part of a convex function of a -dimensional semimartingale = + can be written in terms of an Itô stochastic integral ∫()d, where () is some particular measurable choice of subgradient ∇ f ( x ) of at , and is the martingale part of . This result was first proved by Bouleau in [N. Bouleau, 292 (1981) 87–90]. Here we present a new treatment of the problem. We first prove the result for x10ff65; X = X + ϵB , > 0, where is a standard Brownian...