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Convex function through Doob-Meyer decomposition

Minh Nguyen

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.

Convex function through Doob-Meyer decomposition

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.
Paper Structure (3 sections, 12 theorems, 67 equations)

This paper contains 3 sections, 12 theorems, 67 equations.

Table of Contents

  1. Doob-Meyer decomposition of submartingale
  2. Ito's lemma and stochastic calculus for convex function
  3. From expectation to almost everywhere

Key Result

Lemma 1

For the convex function $f$ with at most polynomial growth, the process $f(W_t^x)$ is a sub-martingale, and the Doob-Meyer decomposition of $f(W_t^x)$ is $f(x) + M^f_t + A^f_t = M_t + A_t$, where $A_t$ is a non-decreasing process and is a positive continuous additive functional (PCAF) of $W$.

Theorems & Definitions (28)

  • Lemma 1
  • proof
  • Definition 1
  • Remark 1
  • Definition 2
  • Definition 3
  • Theorem 1
  • proof
  • Corollary 1
  • Lemma 2
  • ...and 18 more