Summary
In this paper, we characterise the possible joint laws of the maximum and terminal value of a uniformly-integrable martingale. We also characterise the joint laws of the maximum and terminal value of a convergent continuous local martingale vanishing at zero. A number of earlier results on the possible laws of the maximum can be deduced quite easily.
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Rogers, L.C.G. The joint law of the maximum and terminal value of a martingale. Probab. Th. Rel. Fields 95, 451–466 (1993). https://doi.org/10.1007/BF01196729
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DOI: https://doi.org/10.1007/BF01196729