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

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We revisit the celebrated family of BDG-inequalities introduced by Burkholder, Gundy BuGu70 and Davis Da70 for continuous martingales. For the inequalities E[τᵖ/²] ≤ Cₚ E[(B*(τ))ᵖ] with 0 < p < 2 we propose a connection of the optimal constant Cₚ 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.