The First Exit Time of Fractional Brownian Motion with a Drift from a Parabolic Domain

We study the first exit time of a fractional Brownian motion with a drift from a parabolic domain. Actually, we explore three different regimes. In the first regime, the role of drift is negligible. In the second regime, the role of drift is dominating. The behavior of exit probability is the same as that of the crossing probability of a certain moving non-random boundary. In particular, the most interesting, intermediate regime, where all factors come into play, has been solved in this paper. Finally, numerical simulations are conducted, providing an approximate range for the asymptotic estimates to illustrate the practical implications and potential applications of our main results.