Semimartingale decomposition of convex functions of continuous semimartingales by Brownian perturbation

TL;DR

This paper provides a new proof for the semimartingale decomposition of convex functions of continuous semimartingales, using Brownian perturbation and limits, extending Bouleau's earlier result.

Contribution

It introduces a novel approach to semimartingale decomposition by perturbing with Brownian motion and taking limits, offering a different perspective from previous proofs.

Findings

Decomposition expressed via stochastic integral with subgradient

Method using Brownian perturbation and limit process

Extension of Bouleau's original result

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 It^o stochastic integral \int H(X)dM, 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 [2]. Here we present a new treatment of the problem. We first prove the result for X' = X + eB, e > 0, where B is a standard Brownian motion, and then pass to the limit as e tends to 0, using results in [1] and [4].