I'm trying to compute the expected value of a truncated hitting time of a Brownian motion with an one-sided boundary. More specifically, if we let $B_t$ be a $(\mu, \sigma)$ Brownian motion and $T= \inf\{t\ge 0: B_t = a\}$ be a hitting time for some $a>0, I want to know if there is any closed form formula for the expected truncated hitting time

$$ \mathbb{E}[\tau] = \mathbb{E}[T_a \wedge t_0] $$ where $t_0>0$ is a constant. I found out that $T_a$ follows the inverse Gaussian distribution, but I was not able to obtain a closed form solution for the truncated hitting time.


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