A sharp bound on the expected number of upcrossings of an L2-bounded Martingale

For a martingale M starting at x with final variance σ2, and an interval (a,b), let Δ=b−aσ be the normalized length of the interval and let δ=|x−a|σ be the normalized distance from the initial point to the lower endpoint of the interval. The expected number of upcrossings of (a,b) by M is at most 1+δ2−δ2Δ if Δ2≤1+δ2 and at most 11+(Δ+δ)2 otherwise. Both bounds are sharp, attained by Standard Brownian Motion stopped at appropriate stopping times. Both bounds also attain the Doob upper bound on the expected number of upcrossings of (a,b) for submartingales with the corresponding final distribution. Each of these two bounds is at most σ2(b−a), with equality in the first bound for δ=0. The upper bound σ2 on the length covered by M during upcrossings of an interval restricts the possible variability of a martingale in terms of its final variance. This is in the same spirit as the Dubins & Schwarz sharp upper bound σ on the expected maximum of M above x, the Dubins & Schwarz sharp upper bound σ2 on the expected maximal distance of M from x, and the Dubins, Gilat & Meilijson sharp upper bound σ3 on the expected diameter of M.