# Expected value of a Brownian motion before its first hitting time

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

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

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., $$\mathbb{E} \left [ X_{t} | X_{0}=a, H_{b} \right]$$?