medium Pure Math

Hitting MGF

A release-checked medium problem for training Conditional Probability, Conditional Expectation, Stochastic Calculus.

Question

Let $B_t$ be a standard Brownian Motion and define $T_a = \text{inf}\{t > 0: |W_t| > a\}$ for $a > 0$. Find $\mathbb{E}[e^{-\lambda T_a}]$ as a function of $\lambda$ and $a$ and evaulate with: ($\lambda = 4$ and $a = \ln(2)$)

Practice focus

This Pure Math problem is tagged Conditional Probability, Conditional Expectation, Stochastic Calculus. State the random variables and conditioning information explicitly, then check the result against boundary cases before opening hints or a solution.