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

2015/07/28 by Schachermayer, Walter, Stebegg, Florian
#93E20 #FOS: Mathematics #Optimization and Control (math.OC) #Primary 62L15 #Probability (math.PR) #secondary 45J05

paper · doi:10.48550/arxiv.1507.07699

Abstract

We revisit the celebrated family of BDG-inequalities introduced by Burkholder, Gundy \citeBuGu70 and Davis \citeDa70 for continuous martingales. For the inequalities 𝔼[τ^(p)/(2)] ≤ C_p 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 C₁, namely C₁ = 1,27267…. 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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