Let tau_a be the first time standard Brownian motion reaches a > 0. Write the density of tau_a, show that E[tau_a] is infinite, and give the scaling law relating tau_a to tau_1.

Let tau_a be the first time standard Brownian motion reaches a > 0. Write the density of tau_a, show that E[tau_a] is infinite, and give the scaling law relating tau_a to tau_1.

Approach: Differentiate the reflection principle expression for P(tau_a ≤ t) in t, then examine the tail exponent of the resulting density to test integrability.

The density is f(t) = a exp(-a^2/(2t)) / sqrt(2 pi t^3) for t > 0, E[tau_a] = ∞, and tau_a has the law of a^2 tau_1. From the reflection principle P(tau_a ≤ t) = 2(1 - N(a/sqrt t)); differentiating in t gives f(t) = 2 phi(a/sqrt t) * a/(2 t^{3/2}) = a exp(-a^2/(2t))/sqrt(2 pi t^3). For large t the exponential tends to 1, so f(t) ∼ a t^{-3/2}/sqrt(2 pi) and ∫ t f(t) dt behaves like ∫ t^{-1/2} dt, which diverges, so the mean hitting time is infinite even though the level is reached with probability 1. This is the one-sided stable law of index 1/2, and its heavy tail is why an unhedged first passage strategy has unbounded expected holding period. Brownian scaling W_{c^2 t} = c W_t gives tau_a = a^2 tau_1 in law.

Follow-up: What is E[exp(-lambda tau_a)] for lambda > 0, and does it confirm the scaling relation?

Key concepts: first passage time, reflection principle, stable law, heavy tail.