The Brownian Drawdown Process

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 Xt = sups ≤ tXs, the drawdown is thus Xt – Xt, which is a nonnegative process. This is as in figure 1 below.

Brownian motion drawdown
Figure 1: Brownian motion and its drawdown process

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, Xt and |Xt| are identically distributed.

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

\displaystyle  Y_s = X_{t-s} - X_t

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

\displaystyle  Y^*_t = X^*_t-X_t.

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 Yt = Xt – Xt has the same distribution as the absolute value |Yt|= |Xt|. So, the drawdown has the same distribution as the absolute value at each time.

Lemma 1 If X is standard Brownian motion, then Xt – Xt has the same distribution as |Xt| at each time t ≥ 0.

Continue reading “The Brownian Drawdown Process”