The Randomized First-Hitting Problem of Continuously Time-Changed Brownian Motion

Let X (t) be a continuously time-changed Brownian motion starting from a random position h, S (t) a given continuous, increasing boundary, with S (0) >= 0, P (h >= S (0)) = 1, and F an assigned distribution function. We study the inverse first-passage time problem for X (t), which consists in finding the distribution of h such that the first-passage time of X (t) below S (t) has distribution F, generalizing the results, valid in the case when S (t) is a straight line. Some explicit examples are reported.