Hitting probabilities

3.4

What is the probability that a drifting process crosses a barrier by time t — and what does a simulation actually measure?

Goal: compare the continuous-time hitting probability with a discretely monitored simulation, and see how the Broadie–Glasserman–Kou correction closes the gap.

Simulation controls

Level (barrier) a

a1.00

0.22.5

Drift μX(t) = W(t) + μt

μ0.00

-11

Monitoring pointsThe same paths, watched less often

Simulated paths1,000

1002000

Random seed7

19999

Predict first

Monitoring only 100 points, what share of paths is seen crossing a = 1.00?

Your guess

Empirical vs theory

Empirical, 100 points

Empirical, 1600 points

0.284

Continuous theory

0.317

BGK approximation

There are two separate error sources: Monte Carlo noise (shrinks with more paths) and monitoring bias (shrinks only with denser monitoring or a correction). More paths do not remove the bias.

Try this

Predict first: does monitoring at 25 points find more or fewer crossings than at 1600? By how much?
Raise the paths to 2000 at 25 points: does the gap between the empirical and the continuous curve close?
Set a negative drift (μ = −0.5): why does P(τₐ ≤ t) stay well below 1 even for large t?

Fine path and its monitored points

Missed crossing

The fine path (1600 steps) can cross the barrier between two monitoring dates without any monitored point seeing it. That is why discrete monitoring underestimates hitting probabilities.

P(τₐ ≤ t) over time

Empirical (100 points)Continuous theoryBGK-corrected

The discrete empirical curve runs below the continuous theory; the BGK approximation, which shifts the barrier by β√Δt, tracks it.

P(τₐ ≤ 1) as a function of the level

Empirical (100 points)Continuous theoryBGK-corrected

The effect of barrier height at fixed drift. With negative drift the probability does not reach 1 even as t → ∞; it tends to e^(2μa).

3.4