the distribution of the Brownian maximum
Write M_t = max_(0<=u<=t) B_u for the running maximum of Brownian motion over [0, t]. A natural question of physics and finance — how high did the process get, not just where did it end? — is answered exactly by the reflection principle, which gives both the marginal law of M_t and a striking identity tying it to the absolute value |B_t|.
The reflection principle gives, for a >= 0, P(M_t >= a) = 2 P(B_t >= a) = P(|B_t| >= a). Reading off the distribution: M_t has the law of |B_t|, the half-normal (folded normal) distribution. Its density is f_(M_t)(a) = sqrt(2 / (pi t)) exp(-a^2 / (2t)) for a >= 0. Its mean is E[M_t] = sqrt(2t / pi) and variance is (1 - 2/pi) t. More is true: the JOINT law of (M_t, B_t) follows from the same reflection — for a >= max(b, 0) the density is P(M_t in da, B_t in db) = sqrt(2/(pi t^3)) (2a - b) exp(-(2a - b)^2 / (2t)) da db — and from this one recovers the classic identity (M_t, M_t - B_t) has the same law as (|B_t|, L_t) where L_t is local time at 0 (Levy's theorem), i.e. M_t - B_t is distributed like |B_t|.
Why it matters: the maximum's law prices lookback and barrier options, governs maximal deviations in statistics (Kolmogorov-Smirnov), and is the gateway to first-passage times since {M_t >= a} = {tau_a <= t}. A caveat worth stating: M_t equals |B_t| in DISTRIBUTION, not pathwise — M_t is increasing and nonnegative while |B_t| oscillates, so they are very different processes despite sharing a one-dimensional marginal. The deeper pathwise statement is Levy's identity (M_t - B_t, M_t) =d (|B_t|, L_t), which is a theorem about local time, not a triviality.
Over [0, 1], the expected peak of standard Brownian motion is E[M_1] = sqrt(2 / pi) approximately 0.798, even though E[B_1] = 0. The process typically wanders up to roughly four-fifths of a standard deviation above zero before time 1.
The maximum M_t has the half-normal law of |B_t|, with positive mean even though B_t has mean zero.
M_t and |B_t| share a one-dimensional marginal but are NOT equal as processes — M_t is nondecreasing; the genuine pathwise identity is Levy's theorem (M - B, M) =d (|B|, L), involving local time.