Let $B(t)$ be Brownian motion with $B(0)=x>0$ and let $A>x$. It is well known that the expected time for $B(t)$ to escape the interval $[0,A]$ is equal to $x(A-x)$.

Is the expected time known if the process is a fractional Brownian motion with Hurst exponent $H\in (0,1)$ and $H\neq\frac{1}{2}$?


Since martingale methods fail, not much is know about the distribution of hitting times for the fractional Brownian with parameter $H \neq 1/2$. The hitting time of a level is studied by Decreusefond and Nualart in


They get an upper bound for the Laplace transform of the hitting time.

I doubt there is an "explicit" formula for the expected escape time of an interval, but you may be able to obtain an upper bound using the techniques of the above paper.


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