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 ≤ tX_s, the drawdown is thus X^∗_t – X_t, which is a nonnegative process. This is as in figure 1 below.

The previous post used the reflection principle to show that the maximum of a Brownian motion has the same distribution as its terminal absolute value. That is, X^∗_t and |X_t| are identically distributed.

For a process X started from zero, its maximum and drawdown can be written as X^∗_t – X₀ 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.

Continue reading “The Brownian Drawdown Process” →