A note on the maximal expected local time of $${\text {L}}_2$$ -bounded martingales

For an $${\text {L}}_2$$ -bounded martingale starting at 0 and having final variance $$\sigma ^2$$ , the expected local time at $$a \in \text {R}$$ is at most $$\sqrt{\sigma ^2+a^2}-|a|$$ . This sharp bound is attained by Standard Brownian Motion stopped at the first exit time from the interval $$(a-\sqrt{\sigma ^2+a^2},a+\sqrt{\sigma ^2+a^2})$$ . In particular, the maximal expected local time anywhere is at most $$\sigma $$ , and this bound is sharp. 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 (Societé Mathématique de France, Astérisque 157(8), 129–145 1988), by Dubins et al. (Ann Probab 37(1), 393–402 2009) and by the authors (2018).