The Distribution of the Brownian Running Maximum

The Maximum Over a Fixed Time Interval

At a fixed time, standard Brownian motion has a normal distribution. But the probability that it has crossed a level at some point before that time is a different question: a path can cross the level and later finish below it. The reflection principle turns this pathwise event into a calculation involving the endpoint. From it we can obtain the full distribution of the running maximum, not just a single crossing probability.

Let \((\Omega,\mathcal{F},\mathbb{P})\) be a probability space carrying a standard Brownian motion \((B_t)_{t\ge0}\) with continuous paths and \(B_0=0\). Its state space is \(\mathbb{R}\), and its time index is continuous, \(t\ge0\). For a fixed \(T>0\), define the running maximum by \(M_T=\max_{0\le s\le T}B_s\). The maximum exists because each path is continuous on the compact interval \([0,T]\). Since \(B_0=0\), we always have \(M_T\ge0\).

Write \(\Phi(z)=\mathbb{P}(Z\le z)\), where \(Z\) has the standard normal distribution, with density \(\varphi(z)=(2\pi)^{-1/2}e^{-z^2/2}\). Brownian scaling gives \(B_T\) the normal distribution with mean \(0\) and variance \(T\), so \(\mathbb{P}(B_T\ge a)=1-\Phi(a/\sqrt{T})\).

Proof. For \(x>0\), the reflection principle for Brownian motion states that \(\mathbb{P}(M_T\ge x)=2\mathbb{P}(B_T\ge x)\). This result applies to standard Brownian motion started at \(0\), a fixed time \(T>0\), and a positive barrier \(x\). To use it for the cumulative distribution, first note that \(\{M_T>x\}=\bigcup_{n\ge1}\{M_T\ge x+1/n\}\). These events increase as \(n\) increases, so continuity of probability for increasing events and the reflection principle give \(\mathbb{P}(M_T>x)=\lim_{n\to\infty}2\mathbb{P}(B_T\ge x+1/n)=2\mathbb{P}(B_T\ge x)\). The last equality follows from continuity of the normal distribution. Therefore, for \(x>0\), \(\mathbb{P}(M_T\le x)=1-2[1-\Phi(x/\sqrt{T})]=2\Phi(x/\sqrt{T})-1\). For \(x<0\), \(M_T\ge0\) implies \(\mathbb{P}(M_T\le x)=0\). At zero, the events \(\{M_T\le 1/n\}\) decrease to \(\{M_T\le0\}\). The formula for positive \(x\) and continuity of \(\Phi\) show that their probabilities tend to \(2\Phi(0)-1=0\). Thus \(\mathbb{P}(M_T=0)=0\), and the displayed formula also holds at \(x=0\). Finally, the tail formula is continuous in \(x>0\), so \(\mathbb{P}(M_T=x)=\mathbb{P}(M_T\ge x)-\mathbb{P}(M_T>x)=0\) for every \(x>0\). \(\square\)

The key feature is the factor of two. The endpoint event \(\{B_T\ge x\}\) accounts for paths that finish above the barrier; reflection pairs paths that cross the barrier and finish below it with paths that finish above it. Thus the probability that the path reaches the barrier is twice the endpoint’s upper-tail probability. Since \(M_T\ge0\), the resulting distribution has support \([0,\infty)\).

Density, Tails, and Quantiles

Proof. For \(x>0\), differentiate the cumulative distribution from the theorem. The chain rule gives \(\frac{d}{dx}[2\Phi(x/\sqrt{T})-1]=2\varphi(x/\sqrt{T})/\sqrt{T}\), which is the stated density. For \(x<0\) the cumulative distribution is constant, so its density there is zero. The theorem showed that there is no atom at zero. The tail formula follows by subtracting the cumulative distribution from one; equality of the strict and non-strict tails follows because there is no atom at \(a>0\). As a normalization check, integrating the density over \((0,\infty)\) gives \(2\int_0^\infty\varphi(u)\,du=1\), using the change of variable \(u=x/\sqrt{T}\) and symmetry of the standard normal density. Thus the density accounts for the entire probability. \(\square\)

For \(0<p<1\), define the \(p\)-quantile as the smallest \(x\) such that \(F_{M_T}(x)\ge p\). The cumulative distribution is continuous and strictly increasing for \(x\ge0\), because its derivative is positive for \(x>0\). Solving \(2\Phi(x/\sqrt{T})-1=p\) therefore gives \(q_p=\sqrt{T}\,\Phi^{-1}((1+p)/2)\). The inverse is evaluated at a number strictly between \(1/2\) and \(1\), so this quantile is positive and finite. The endpoints require care: the lower endpoint of the support is \(0\), while no finite \(x\) has \(F_{M_T}(x)=1\).

Scaling and a Common Pitfall

The distribution also makes the time scaling transparent. For \(c>0\), the density of \(M_T\) is the density of \(\sqrt{T}M_1\): substituting \(x=\sqrt{T}y\) into the density at time \(T\) gives \(\sqrt{2/\pi}\,e^{-y^2/2}\) as the density in \(y\). Equivalently, the quantile at any fixed probability \(p\) is \(\sqrt{T}\) times the corresponding quantile at time \(1\). A longer time interval increases the typical size of the maximum on the square-root time scale.

Common mistake: using the endpoint tail without the reflection factor. At \(T=4\) and \(a=2\), the endpoint probability is \(\mathbb{P}(B_4\ge2)=1-\Phi(1)\approx0.1586553\). That is not the probability that the path reaches \(2\): it misses paths that cross \(2\) and then end below it. The reflection principle gives the correct result \(2[1-\Phi(1)]\approx0.3173106\). In general, the endpoint tail is the right-hand side of the reflection identity before multiplication by \(2\), not the running-maximum tail itself.

These formulas apply to the maximum of \(B_s\), not the maximum of \(|B_s|\), and they describe the maximum over the whole interval rather than the endpoint alone. Keeping those random quantities distinct is essential: the running maximum records the entire path, whereas \(B_T\) records only its final value.