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This fact is essential for the proof of the existence of the Doob-Meyer decomposition for square-integrable martingales with a two dimensional parameter set
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Nguyen-Minh-Duc, Nguyen-Xuan-Loc On the transformation of martingales with a two dimensional parameter set by convex functions. Z. Wahrscheinlichkeitstheorie verw Gebiete 66, 19–24 (1984). https://doi.org/10.1007/BF00532793
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DOI: https://doi.org/10.1007/BF00532793