An inverse first-passage problem revisited: the case of fractional Brownian motion, and time-changed Brownian motion

We revisit an inverse first-passage time (IFPT) problem, in the cases of fractional Brownian motion, and time-changed Brownian motion. Let $X(t)$ be a one dimensional continuous stochastic process starting from a random position $eta$ , let $S(t)$ be an assigned continuous boundary, such that $ eta ge S(0)$ with probability one, and F an assigned distribution function. The IFPT problem here considered consists in finding the distribution of $eta$ such that the first-passage time of X(t) below S(t) has distribution F. We study this IFPT problem for fractional Brownian motion and a constant boundary $S(t)=S$; we also obtain some extension to other Gaussian processes, for one, or two, time dependent boundaries.