Convex function through Doob-Meyer decomposition

TL;DR

This paper develops a probabilistic approach to a strong version of Ito's lemma for convex functions using Doob-Meyer decomposition, providing new insights into their second-order differentiability.

Contribution

It introduces a novel probabilistic proof of Ito's lemma for convex functions via Doob-Meyer decomposition, linking stochastic calculus with convex analysis.

Findings

Establishes a strong Ito's lemma for convex functions using probabilistic methods.

Shows convex functions are second-order differentiable through stochastic inequalities.

Provides a new perspective connecting stochastic calculus and convex analysis.

Abstract

In this work, we aim to study a strong version of Ito's lemma for convex function. By considering the corresponding sub-martingale on a Brownian motion, we gain more insights about the convex function through a probabilistic viewpoint. The Doob-Meyer decomposition of this sub-martingale subsequently helps us deduce the Ito's lemma for convex function, and enables us to study a convex function via stochastic calculus. In particular, we use this version of Ito's lemma together probabilistic inequalities to recover an important analytic property of the convex function, which is its second-order differentiability.