Let $(B)_{t\geq 0}$ be a standard Brownian motion and let $c\in\mathbb{R}$. Can we calculate the expectation of the exponential of the time $B$ will be spent above $c$, $$\mathbb{E}\left[\exp\left\{\int_0^t\mathbb{1}_{\{B_u>c\}}du\right\}\right]?$$
My thought is: Let $L_t:=\exp\left\{\int_0^t\mathbb{1}_{\{B_u>c\}}du\right\}$, it is a non-decreasing process, hence, a sub-martingale. Next step is to identify its Doob-Meyer decomposition, and this is where I cannot proceed.
