Energy of taut strings accompanying Wiener process

Let $W$ be a Wiener process. The function $h(\cdot)$ minmizing energy $\int_0^T h'(t)^2\, dt$ among all functions satisfying $W(t)-r \le h(t) \le W(t)+ r$ on an interval $[0,T]$ is called taut string. This is a classical object well known in Variational Calculus, Mathematical Statistics, etc. We show that the energy of this taut string on large intervals is equivalent to $C^2 T\, /\, r^2$ where $C$ is some finite positive constant. While the precise value of $C$ remains unknown, we give various theoretical bounds for it as well as rather precise results of computer simulation. While the taut string clearly depends on entire trajectory of $W$, we also consider an adaptive version of the problem by giving a construction (Markovian pursuit) of a random function based only on the past values of $W$ and having minimal asymptotic energy. The solution, an optimal pursuit strategy, quite surprisingly turns out to be related with a classical minimization problem for Fisher information on the bounded interval.