Let $X_{t}$ be a Brownian motion with drift $\mu=0$ and variance $\sigma^{2}$. Also, let $X_{0} = a < b$. We know that the density of the first hitting time $H_{b} = inf \lbrace t: X_{t} = b \rbrace$ is

\begin{equation} f_{H_{b}} = \frac{ (b-a) e^{-\frac{(b-a)^{2}}{2t}}}{\sqrt{2\pi t^{3}}} \end{equation}

Hence, my question is: what is the expected value of $X_{t}$, for $0 \leqslant t \leqslant H_{b}$, given that $X_{0} = a$ and the first hitting time at point $b$ is $H_{b}$, i.e., $\mathbb{E} \left [ X_{t} | X_{0}=a, H_{b} \right] $?


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