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November 2018 The sharp constant for the Burkholder–Davis–Gundy inequality and non-smooth pasting
Walter Schachermayer, Florian Stebegg
Bernoulli 24(4A): 2499-2530 (November 2018). DOI: 10.3150/17-BEJ935

Abstract

We revisit the celebrated family of BDG-inequalities introduced by Burkholder, Gundy (Acta Math. 124 (1970) 249–304) and Davis (Israel J. Math. 8 (1970) 187–190) 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 $C_{1}=1.27267\ldots$ . 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. "The sharp constant for the Burkholder–Davis–Gundy inequality and non-smooth pasting." Bernoulli 24 (4A) 2499 - 2530, November 2018. https://doi.org/10.3150/17-BEJ935

Information

Received: 1 January 2016; Revised: 1 September 2016; Published: November 2018
First available in Project Euclid: 26 March 2018

zbMATH: 06853256
MathSciNet: MR3779693
Digital Object Identifier: 10.3150/17-BEJ935

Rights: Copyright © 2018 Bernoulli Society for Mathematical Statistics and Probability

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Vol.24 • No. 4A • November 2018
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