Energy of taut strings accompanying Wiener process

Mikhail Lifshits, Eric Setterqvist

Let $W$ be a Wiener process. The function $h(\cdot )$ minmizing energy ${\int }_0^T{h}^{\prime }(t{)}^{2}dt$ among all functions satisfying $W(t)-r\le h(t)\le W(t)+r$ on an interval $[0,T]$ is…