A general downcrossing inequality for g-martingales

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Dual Formulation of Second Order Target Problems (2010, methodology): “Following the definition in [15] and [4], Ṽ P is a weak f -supermartingale under P.”
Optimal stopping under ambiguity in continuous time (2013, methodology): “The Optional Sampling Theorem of Chen and Peng (2000) show that strong and weak g–supermartingales coincide.”
Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation (2006, unknown): “Since then many results have been obtained in this subject (see, among others, [3], [4], [5], [6], [10], [11], [7], [8], [24], [25], [35], [38], [39], [41], [45]).”
NONLINEAR EXPECTATIONS AND NONLINEAR MARKOV CHAINS (2005, unknown): “[BCHMP2000[2]], [Chen1998[3]], [ChenPeng2001[9]], [Chen-Peng2000[10]].”
Filtration-consistent nonlinear expectations and related g-expectations (2002, methodology): “Contrarily to Theorem 6 in [5], we need not assume that Y is positive: indeed, as g(·, y, 0) = 0, one checks easily that the proof given in [5] can be carried over for every g-supermartingale.”