Integral Inequalities for Convex Functions of Operators on Martingales

By Burgess Davis, Renming Song. 2011

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On the existence and position of the farthest peaks of a family of stochastic heat and wave equations (2010, unknown): “4 (The Burkholder–Davis–Gundy inequality [2, 3, 4]).”
Extragradient Method with Variance Reduction for Stochastic Variational Inequalities (2017, background): “ < ∞ and P P bk < ∞. Then, a.s. {yk} converges and uk < ∞. If a.s. for all k ∈ N, E[yk+1|Fk] = yk then {yk,Fk} is called a martingale. We shall also need the following moment inequality; see [4, 28]. Theorem 2 (The inequality of Burkholder-Davis-Gundy). We denote as k · k the Euclidean norm”
Antithetic multilevel Monte Carlo estimation for multi-dimensional SDEs without Lévy area simulation (2012, methodology): “2) involve martingales, and so we can use the discrete Burkholder-Davis-Gundy inequality [1].”
Discretely sampled signals and the rough Hoff process (2013, background): “In the case of x ∈ C ( [0, 1],R ) where p ∈ [1, 2), the iterated integrals of x are canonically defined via Young integration [31].”