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.

  • 1
    $\begingroup$ Borodin, Salminen, P. 128, formula 1.4.3. $\endgroup$
    – zhoraster
    Sep 16, 2022 at 5:06
  • $\begingroup$ From the answer by user zhoraster, the result is given by formula 1.4.3 Borodin and Salminen. $\endgroup$
    – Amira
    Sep 20, 2022 at 15:56


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