A note on the maximal expected local time of L^2 -bounded martingales

For a continuous L2-bounded Martingale with no intervals of constancy, starting at 0 and having final variance σ2, the expected local time at x∈R is at most σ2+x2−−−−−−√−|x|. This sharp bound is attained by Standard Brownian Motion stopped at the first exit time from the interval (x−σ2+x2−−−−−−√,x+σ2+x2−−−−−−√). Sharp bounds for the expected maximum, maximal absolute value, maximal diameter and maximal number of upcrossings of intervals, have been established by Dubins and Schwarz (1988), Dubins, Gilat and Meilijson (2009) and by the authors (2017).