Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 202 / 3 / b / Solution

The exponential process $Z_t=\exp (-B_t-t/2)$ is a martingale. Under the Girsanov theorem change of measure $d\mathbb{Q}{|}_{{\mathcal{F}}_t}=Z_t,d\mathbb{P}{|}_{{\mathcal{F}}_t}$, the process $W_t=B_t+t$ is Brownian motion. Its first hitting time $T_a$ of $a>0$ is finite $\mathbb{Q}$-almost surely. On $\{T_a\le t\}$, the optional sampling theorem givesbecause $B_{T_a}=a-T_a$. Letting $t\to \infty$ and applying the monotone convergence theorem yieldsThis is the Critical exponential moment of a drifted Brownian hitting time.