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

Abstract

We study an inverse first-passage-time problem for Brownian motion $X(t),$ starting from a fixed point x. For $t\ge 0,$ let be $S(t)=A+bt$ a randomly perturbed straight line, where $A=S(0)$ is a random variable, independent of x, such that $A\le x,$ while $b\ge 0$ 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 $H=1,$ and b = 0 is considered. Some explicit examples are reported.

Keywords:

• Brownian motion
• first-passage time
• inverse first-passage-time problem

MATHEMATICS SUBJECT CLASSIFICATION:

• 60J60
• 60H05
• 60H10

Additional information

Funding

The author acknowledges the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006.