Summary
Given any local maringaleM inR d orl 2, there exists a local martingaleN inR 2, such that |M|=|N|, [M]=[N], and «M»=«N». It follows in particular that any inequality for martingales inR 2 which involves only the processes |M|, [M] and «M» remains true in arbitrary dimension. WhenM is continuous, the processes |M|2 and |M| satisfy certain SDE's which are independent of dimension and yield information about the growth rate ofM. This leads in particular to tail estimates of the same order as in one dimension. The paper concludes with some new maximal inequalities in continuous time.
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Research supported by NSF grant DMS-9002732 and by AFOSR Contract F49620 85C 0144
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Kallenberg, O., Sztencel, R. Some dimension-free features of vector-valued martingales. Probab. Th. Rel. Fields 88, 215–247 (1991). https://doi.org/10.1007/BF01212560
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DOI: https://doi.org/10.1007/BF01212560