An inverse first-passage problem revisited: the case of fractional Brownian motion, and time-changed Brownian motion

Abstract

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 $\eta ,$ and let S(t) be an assigned continuous boundary, such that $P(\eta \ge S(0))=1,$ and F an assigned distribution function. The IFPT problem here considered consists in finding the distribution of η 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 $S(t)=S;$ we also obtain some extension to other Gaussian processes, for one, or two, time-dependent boundaries.

Keywords:

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

Mathematics Subject Classification:

• 60J60
• 60H05
• 60H10

Acknowledgments

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