Let $X$ be a geometric Brownian motion, satisfying the SDE

$$dX_t = \sigma X_t \, dW_t, X_0 = 1.$$

for $W$ a standard one dimensional Brownian motion, and $\sigma > 0$ a constant.

Define the *drawdown process* $D$ by

$$D_t := \max_{0 \leq s \leq t} X_s - X_t.$$

and for some fixed $a >0$, let $\tau$ be the stopping time

$$\tau := \inf \{t > 0 \, | \, D_t =a\}.$$

**Question:** Can we obtain, or characterise the joint distribution of $(\tau, X_\tau)$?