Classify survival probability in the borderline law-of-iterated-logarithm regime

Determine whether the survival probability P(τ < ∞) equals 1 or is strictly less than 1 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 positive barrier B(t) with B(0) = K < V_0, under the simultaneous conditions: (i) limsup_{t→∞} B(t) / [K exp((μ − σ^2/2) t − σ sqrt(2 t ln ln t))] ≥ 1, (ii) liminf_{t→∞} B(t) / [K exp((μ − σ^2/2) t − σ sqrt(2 t ln ln t))] < 1, and (iii) limsup_{t→∞} B(t) / [K exp((μ − σ^2/2) t + σ sqrt(2 t ln ln t))] ≤ 1, in order to resolve the classification between P(τ < ∞) = 1 and 0 < P(τ < ∞) < 1.

Background

The paper studies first passage times of geometric Brownian motion through general continuous barriers and classifies default risk into regimes based on whether the first passage time is almost surely finite and whether its mean is finite. Using the law of the iterated logarithm, the authors establish criteria guaranteeing P(τ < ∞) = 1 and criteria ensuring 0 < P(τ < ∞) < 1.

However, when the barrier’s long-time behavior oscillates near the critical law-of-iterated-logarithm envelopes, the established results do not decide whether survival to infinity has positive probability or is null. Specifically, if the barrier’s limsup relative to the lower LIL envelope is at least one, its liminf is below one, and its limsup relative to the upper LIL envelope is at most one, the current theorems do not determine whether P(τ < ∞) equals 1 or lies strictly between 0 and 1. The authors explicitly note this undecidability and call for additional conditions to complete the classification.