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

We study an inverse first-passage-time problem for Brownian motion XðtÞ, starting from a fixed point x. For t 0, let be SðtÞ ¼ A þ bt a randomly perturbed straight line, where A ¼ Sð0Þ is a random variable, independent of x, such that A x, while b 0 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 H ¼ 1, and b ¼ 0 is considered. Some explicit examples are reported.