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 and let S(t) be an assigned continuous boundary, such that 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 we also obtain some extension to other Gaussian processes, for one, or two, time-dependent boundaries.