For standard Brownian motion with running maximum M_t, prove that P(M_t ≥ a, W_t ≤ b) = P(W_t ≥ 2a - b) for b ≤ a and a > 0, and use it to write the joint density of (M_t, W_t).

For standard Brownian motion with running maximum M_t, prove that P(M_t ≥ a, W_t ≤ b) = P(W_t ≥ 2a - b) for b ≤ a and a > 0, and use it to write the joint density of (M_t, W_t).

Approach: Reflect the path in the level a after the first passage time, then differentiate the resulting tail probability once in each argument.

P(M_t ≥ a, W_t ≤ b) = P(W_t ≥ 2a - b) = 1 - N((2a - b)/sqrt(t)), and the joint density of the running maximum and the terminal value is f(m,w) = sqrt(2/(pi t^3)) (2m - w) exp(-(2m - w)^2/(2t)) on m ≥ max(0,w). On the event that M_t ≥ a the first passage time tau_a is at most t, and the strong Markov property makes W_{tau_a + u} - a a fresh Brownian motion, so reflecting that piece in the level a is measure preserving. Reflection maps the event {tau_a ≤ t, W_t ≤ b} onto {tau_a ≤ t, W_t ≥ 2a - b}, and since 2a - b ≥ a the passage condition is automatic on the image, which gives the identity. Differentiating with respect to b and then with respect to a, with attention to the two sign changes, yields the stated density.

Follow-up: What is the law of M_t - W_t, and which process does M - W turn out to be?

Key concepts: reflection principle, running maximum, joint density, strong Markov property.