Joint distribution of drawdown time and value of geometric Brownian motion

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 $$(\tau, X_\tau)$$?

Before you get no response i think i can provide you with some limited insight into the distribution. Firstly with $$X_0 = 1 \implies X_\tau \leq a+1$$ and since $$X_t > 0, \forall t > 0$$, we can also say that $$X_\tau > a$$.
From these two observations we can see that $$\tau$$ is bounded by two other stoppage times: $$\tau_- = \inf\{t > 0|X_t = a \} \leq \tau \leq \inf\{t>0 |X_t = a +1 \} = \tau_+$$
Now $$\tau_{-, +}$$ can be transformed to be about $$W_t$$, as in $$\tau_- = \inf\{t>0|W_t = \frac{\log(a)}{\sigma}\}$$, $$\tau_+ = ...$$. Professor Steve Lalley calls this a first-passage time in this course: http://galton.uchicago.edu/~lalley/Courses/312/ (relevant subsections, 2nd Random walk and last Brownian Motion)
The expectation value of such passage times for Wiener Processes are $$\infty$$ because they are Martingales and this a type of Gamblers Ruin: https://en.wikipedia.org/wiki/Optional_stopping_theorem#Applications
So that's pretty much it. $$X_\tau \in (a, a+1]$$ and $$\mathbb E(\tau) = \infty$$. I hope it helps.