The drawdown of a stochastic process is the amount that it has dropped since it last hit its maximum value so far. For process X with running maximum X^{∗}_{t} = sup_{s ≤ t}X_{s}, the drawdown is thus X^{∗}_{t} – X_{t}, which is a nonnegative process. This is as in figure 1 below.

For a process X started from zero, its maximum and drawdown can be written as X^{∗}_{t} – X_{0} and X^{∗}_{t} – X_{t}. Reversing the process in time across the interval [0, t] will exchange these values. So, reversing in time and translating so that it still starts from zero will exchange the maximum value and the drawdown. Specifically, write

for time index 0 ≤ s ≤ t. The maximum of Y is equal to the drawdown of X,

If X is standard Brownian motion then so is Y, since the independent normal increments property for Y follows from that of X. As already stated, the maximum Y^{∗}_{t} = X^{∗}_{t} – X_{t} has the same distribution as the absolute value |Y_{t}|= |X_{t}|. So, the drawdown has the same distribution as the absolute value at each time.

Lemma 1 If X is standard Brownian motion, then X^{∗}_{t} – X_{t} has the same distribution as |X_{t}| at each time t ≥ 0.