New and refined bounds for expected maxima of fractional Brownian motion

Abstract

For the fractional Brownian motion B H with the Hurst parameter value H in (0,1/2), we derive new upper and lower bounds for the difference between the expectations of the maximum of B H over [0,1] and the maximum of B H over the discrete set of values in −1 , i = 1, . . . , n. We use these results to improve our earlier upper bounds for the expectation of the maximum of B H over [0, 1] and derive new upper bounds for Pickands' constant.

Derivatives of sup-functionals of fractional Brownian motion evaluated at H=1/2

“…Thus, most of the work is focused on finding upper and lower bounds for these quantities (see, e.g. [35,12,10,4,7,8,5]) or determining their asymptotic behavior in various settings (as H goes to 0, H goes to 1, T grows large, a goes to 0 etc. ); see, e.g., [21,5].…”

On the speed of convergence of discrete Pickands constants to continuous ones

“…A problem related to the asymptotic behavior of H α − H δ α was considered in [6,7], who have shown that E sup t∈[0,1] B α (t) − E sup t∈[0,1] δ B α (t), decays like δ α/2 up to logarithmic terms. We should emphasize that in Theorem 2.1, case α ∈ (0, 1) we were able to establish that the upper bound for the discretization error decays exactly like δ α/2 .…”

Lower bound for the expected supremum of fractional Brownian motion using coupling

“…Since the estimation of M H (T, a) is so challenging, many works are dedicated to finding its theoretical upper and lower bounds. The most up-to-date bounds for M H (T, 0) can be found in [8,9], see also [23,26] for older results. The most up-to-date bounds for M H (∞, a) can be found in [4].…”