Randomization of a linear boundary in the first-passage problem of Brownian motion

We study an inverse first-passage-time problem for Brownian motion starting from a fixed point x. For let be a randomly perturbed straight line, where is a random variable, independent of x, such that while is fixed, and let be F an assigned distribution function. The problem consists in finding the distribution of A such that the first-passage time of X(t) below S(t) has distribution F. The analogous case for fractional Brownian motion with Hurst index and b = 0 is considered. Some explicit examples are reported.