2023/10/03 by Dmitriy Stolyarov, Stolyarov, Dmitriy, Vasily Vasyunin +3
Economics, Econometrics and Finance · #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2310.02362
We provide sharp estimates for the distribution function of a martingale transform of the indicator function of an event. They are formulated in terms of Burkholder functions, which are reduced to the already known Bellman functions for extremal problems on BMO. The reduction implicitly uses an unexpected phenomenon of automatic concavity for those Bellman functions: their concavity in some directions implies concavity with respect to other directions. A similar question for a martingale transform of a bounded random variable is also considered.