Let $B$ be a standard Brownian motion and let $S$ be a random variable independent of $B$ with the standard exponential distribution (so, $\mathbb{P}\{S > t \} = \exp(-t)$ for $t \ge 0$), and let $\theta > 0$ be a constant.

1. Write down an explicit integral formula for the probability density of $B_{2S/\theta^2}$ by conditioning on $S$.

2. Let $T_x = \inf\{t > 0 \;:\; B_t = x\}$ be the first hitting time of $x$. Use the reflection principle and a Brownian scaling argument to show that $T_x \stackrel{\scriptscriptstyle{d}}{=} (x / B_1)^2$.

3. Recall that $\mathbb{E}[e^{-\lambda T_x}] = \exp(-x \sqrt{2\lambda})$, recognize that this formula is involved in the integral formula from part (1), and use this to simplify the integral.

4. Conclude from the simplification that \(B_{2S/\theta^2} \stackrel{\scriptscriptstyle{d}}{=} (S - S')/\theta,\) where $S$ and $S’$ are independent copies of $S$.

Attribution: This problem is modified from worksheets provided by Jim Pitman (thanks!).