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arXiv · 1507.07699

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

Walter Schachermayer·Florian Stebegg
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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 $\mathbb{E}[τ^{\frac{p}{2}}] \leq C_p \mathbb{E}[(B^*(τ))^p]$ 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.

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Walter Schachermayer, Florian Stebegg. 2017-03-03. The Sharp Constant for the Burkholder-Davis-Gundy Inequality and Non-Smooth Pasting. https://arxiv.org/abs/1507.07699

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