Energy of taut strings accompanying Wiener process

Let W be a Wiener process. For r>0 and T>0 let IW(T,r)2 denote the minimal value of the energy ∫0Th′(t)2dt taken among all absolutely continuous functions h(⋅) defined on [0,T], starting at zero and satisfying W(t)−r≤h(t)≤W(t)+r,0≤t≤T. The function minimizing energy is a taut string, a classical object well known in Variational Calculus, in Mathematical Statistics, and in a broad range of applications. We show that there exists a constant C∈(0,∞) such that for any q>0rT1/2IW(T,r)⟶LqC,as  rT1/2→0, and for any fixed r>0, rT1/2IW(T,r)⟶a.s.C,as  T→∞. Although precise value of C remains unknown, we give various theoretical bounds for it, as well as rather precise results of computer simulation.