Decide finiteness of the mean first passage time at the critical square-root barrier

Ascertain whether the mean first passage time E[τ] is finite or infinite for the first passage time τ of the geometric Brownian motion V_t = V_0 exp((μ − σ^2/2) t + σ W_t) through a continuous barrier B(t) when B(t) oscillates around or asymptotically equals the critical barrier B_c(t, σ) = K exp((μ − σ^2/2) t + σ sqrt(t)), namely in the cases where (a) liminf_{t→∞} B(t)/B_c(t, σ) < 1 and limsup_{t→∞} B(t)/B_c(t, σ) > 1 both hold, or (b) lim_{t→∞} B(t)/B_c(t, σ) = 1 but B(t) is not uniformly bounded above by B_c(t, σ).

Background

The authors identify a critical square-root barrier B_c(t, α) = K exp((μ − σ^2/2) t + α sqrt(t)) delineating the transition between finite and infinite mean first passage times: if B(t) eventually lies above B_c(t, α) with α > σ, then E[τ] < ∞; if B(t) eventually lies below B_c(t, σ), then E[τ] = ∞.

They explicitly note that their classification does not resolve borderline regimes where the barrier oscillates around the critical envelope or converges to it, leaving the finiteness of E[τ] undecided in those cases. Determining the mean behavior in these limit configurations would close a gap in the qualitative risk classification framework.