First passage times

3.3

When does a Brownian path first reach level a, and how long do we have to wait?

Goal: stop paths at the level, measure the distribution of first passage times and compare it with the Lévy density.

Simulation controls

Level (barrier) aDoubling a roughly quadruples the waiting time: τₐ ∼ a²·τ₁

a0.50

0.22.5

Simulated paths1,500

1003000

Random seed7

19999

Predict first

What share of paths touches the barrier a = 0.50 at least once by t = 1?

Your guess

Empirical vs theory

P(τₐ ≤ 1) empirical

2(1 − Φ(a)) theory

Median of τₐ, empirical

0.672

Median of τₐ, theory

0.550

Almost every path eventually reaches level a, yet E[τₐ] = ∞: rare, very long waits weigh enough to make the mean infinite. That is why the median is the meaningful summary here.

Calibration: 90% range

Give a 90% range for the first time the path reaches a = 0.50.

The hitting-time law is heavy-tailed: its mean is infinite, so allow a long right tail.

Your 90% range: 0.10 … 1.00

Try this

Predict first: if you double a, by what factor does the median of τₐ grow? Check a = 0.5 against a = 1.
Look at the density near t ≈ 0: why is an almost immediate hit practically impossible?
Push a above 2: what happens to the fraction of paths that hit within [0, 1] and to the area under the histogram?

Paths stopped at the level, W(t ∧ τₐ)

a = 0.50

Each path freezes as soon as it reaches level a. The fraction collecting at the level grows with t, but slowly: many paths take a very long time to get there.

Density of τₐ on [0, 1]

τₐ empiricalLévy density

The histogram is normalised by the full sample, so its area is P(τₐ ≤ 1) < 1 — the remaining paths arrive after t = 1. The curve is a/√(2πt³)·exp(−a²/2t).

3.3