W is standard Brownian motion. Compute P(max_{0 ≤ s ≤ 1} W_s ≥ 2) to four decimal places, and state the general formula for the running maximum.

W is standard Brownian motion. Compute P(max_{0 ≤ s ≤ 1} W_s ≥ 2) to four decimal places, and state the general formula for the running maximum.

Approach: Use the reflection principle to relate the event that the maximum exceeds a level to the terminal value crossing that level, then read the answer off the normal cdf.

0.0455. The reflection principle says that for a > 0, P(M_t ≥ a) = 2 P(W_t ≥ a), because every path whose first passage to a occurs before t splits into two equally likely continuations after that time, one ending above a and one ending below. Here 2 P(W_1 ≥ 2) = 2(1 - N(2)) = 2(0.02275) = 0.0455. The general statement for the running maximum, in terms of the normal cdf N, is P(M_t ≥ a) = 2(1 - N(a / sqrt(t))), so M_t has the same distribution as |W_t|, with density 2 phi(a / sqrt t) / sqrt t on a > 0. The same computation gives the first passage law P(tau_a ≤ t) = P(M_t ≥ a).

Follow-up: What is P(max_{0 ≤ s ≤ 1} W_s ≥ 2 and W_1 ≤ 0), and how does it compare to the unconditional 0.0455?

Key concepts: reflection principle, running maximum, normal cdf, first passage.