Let $\beta > 0$, $0 < \gamma < 1$, and let $\tau$ be the first hitting time: $$\tau = \inf\{t:t \geq 0, |W_t| = \pi /4\}$$ Solve the SDE in the random interval $0 \leq t \leq \tau$ $$dX_t = -(\beta + \dfrac{1}{2} \gamma^2)X_t dt + \gamma \sqrt{e^{-2 \beta t} -X_t^2} dW_t$$ $$ X_0 = \dfrac{\sqrt{2}}{2}$$
Remarks: I believe the question is begging for some symmetry exploitation. Indeed, for fixed $\tau$, $W_{\tau}$ takes value $\pi/4$ w.p. 1/2 and $-\pi/4$ w.p. $1/2$. But I am having trouble understanding the connection between this interval, stopping times, and the solution process to a PDE.
My professor has only taught us multiplicative factor technique for solving SDE. I don't know how to deal with the random interval here. How does one even start this problem? I believe there is a trick which makes the solution short. Which techniques should I start with?