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Let $W$ be a Wiener process. The function $h(\cdot)$ minmizing energy$\int_0^T h'(t)^2\, dt$ among all functions satisfying $W(t)-r \le h(t) \leW(t)+ r$ on an interval $[0,T]$ is called taut string. This is a classicalobject well known in Variational Calculus, Mathematical Statistics, etc. Weshow 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 precisevalue of $C$ remains unknown, we give various theoretical bounds for it as wellas rather precise results of computer simulation.  While the taut string clearly depends on entire trajectory of $W$, we alsoconsider an adaptive version of the problem by giving a construction (Markovianpursuit) of a random function based only on the past values of $W$ and havingminimal asymptotic energy. The solution, an optimal pursuit strategy, quitesurprisingly turns out to be related with a classical minimization problem forFisher information on the bounded interval.

Energy of taut strings accompanying Wiener process