## Bernoulli

• Bernoulli
• Volume 24, Number 4A (2018), 2499-2530.

### The sharp constant for the Burkholder–Davis–Gundy inequality and non-smooth pasting

#### 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.

#### Article information

Source
Bernoulli, Volume 24, Number 4A (2018), 2499-2530.

Dates
Revised: September 2016
First available in Project Euclid: 26 March 2018

https://projecteuclid.org/euclid.bj/1522051216

Digital Object Identifier
doi:10.3150/17-BEJ935

Mathematical Reviews number (MathSciNet)
MR3779693

Zentralblatt MATH identifier
06853256

#### Citation

Schachermayer, Walter; Stebegg, Florian. The sharp constant for the Burkholder–Davis–Gundy inequality and non-smooth pasting. Bernoulli 24 (2018), no. 4A, 2499--2530. doi:10.3150/17-BEJ935. https://projecteuclid.org/euclid.bj/1522051216

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