Let $(B_t)_{t\geq 0}$ be a standard Brownian motion and $c\in\mathbb{R}$ be a constant. Calculate the following expectation: $$\mathbb{E}\left[\exp\left\{-\int_0^t\mathbb{1}_{\{B_u>c\}}du-\frac{1}{2}t+B_t\right\}\right].$$
Observe that $e^{-\frac{1}{2}t+B_t}$ is a martingale, however, due to the dependence with the term $\int_0^t\mathbb{1}_{\{B_u>c\}}du$ (which is the time of B spent above the barrier $c$), we cannot directly take the advantage of the martingale property. Moreover, the expectation of $\mathbb{E}\left[\exp\left\{-\int_0^t\mathbb{1}_{\{B_u>c\}}du\right\}\right]$ is explicit and is given by formula 1.4.3 Borodin and Salminen.