The Sharp Constant for the Burkholder-Davis-Gundy Inequality and Non-Smooth Pasting

TL;DR

This paper investigates the optimal constants in Burkholder-Davis-Gundy inequalities for continuous martingales, linking them to integro-differential equations and discovering a phenomenon called non-smooth pasting.

Contribution

It introduces a novel connection between the BDG constants and integro-differential equations, enabling numerical computation of these constants, including an explicit value for p=1.

Findings

Explicit value of C_1 ≈ 1.27267 obtained numerically.

Identification of non-smooth pasting in solutions of related equations.

Development of a numerical method for optimal constant calculation.

Abstract

We revisit the celebrated family of BDG-inequalities introduced by Burkholder, Gundy \cite{BuGu70} and Davis \cite{Da70} for continuous martingales. For the inequalities $\mathbb{E}[\tau^{\frac{p}{2}}] \leq C_p \mathbb{E}[(B^*(\tau))^p]$ with $0 < p < 2$ we propose a connection of the optimal constant $C_p$ with an ordinary integro-differential equation which gives rise to a numerical method of finding this constant. Based on numerical evidence we are able to calculate, for $p=1$, the explicit value of the optimal constant $C_1$, namely $C_1 = 1,27267\dots$. In the course of our analysis, we find a remarkable appearance of "non-smooth pasting" for a solution of a related ordinary integro-differential equation.