Bounds for expected supremum of fractional Brownian motion with drift

We provide upper and lower bounds for the mean M (H) of supp t≥0{BH (t)} , with BH (.) a zero-mean, variance-normalized version of fractional Brownian motion with Hurst parameter H ∈ (o,1). We find bounds in (semi-) closed form, distinguishing between H ∈ (0, ½] and H ∈ [½, 1) , where in the former regime a numerical procedure is presented that drastically reduces the upper bound. For H ∈ (0, ½] , the ratio between the upper and lower bound is bounded, whereas for H ∈ [½, 1) the derived upper and lower bound have a strongly similar shape. We also derive a new upper bound for the mean of sup t∈[0,1] BH (t), H ∈ (0, ½] , which is tight around H = ½