First-Passage Probabilities and Optimal Barriers for Brownian Motion with Drift

Let $X_t = \mu t + W_t$ where $W_t$ is a standard Brownian motion and $\mu$ is a constant drift. Two absorbing barriers sit at $-L < 0$ and $U > 0$. Define $\tau = \inf\{t \geq 0 : X_t \in \{-L, U\}\}$ as the first time the process hits either barrier. 1. Compute $P(X_\tau = U)$, the probability of hitting the upper barrier before the lower. 2. Compute $E[\tau]$, the expected time to absorption. 3. Describe what happens in the limiting cases $\mu \to 0$ and $U, L \to \infty$. 4. Suppose you can choose a single symmetric barrier $b = U = L > 0$ before trading begins. How should the optimal $b$ change with $|\mu|$ to balance hit probability against expected duration? Discuss the trade-off qualitatively and describe what the optimal $b$ looks like as $|\mu| \to 0$ and $|\mu| \to \infty$.