(1) Deduce from Proposition 2.5.4.1 P ( A , 0 t d u W...

(1) Deduce from Proposition 2.5.4.1 $P(A_t^{-,0}\in du\mid W_t=x)$.

(2) Recover the formula (2.5.5) for $P(A_t^{-,0}\in du)$.

Proposition 2.5.4.1:

The density of the pair $(A_t^{-,0},W_t)$ is

$$P(A_t^{-,0}\in du,W_t\in dx)=dudx\frac{|x|}{\sqrt{2\pi }}{\int }_u^t\frac{1}{\sqrt{s^3(t-s{)}^3}}{e}^{-x^2/\left(2\right(t-s\left)\right)}ds{1}_{\{u<t\}}$$

$$\begin{array}{c}P(A_t^{-,0,\nu }\in du)/du=[\sqrt{\frac{2}{\pi u}}\exp (-\frac{{\nu }^2}{2}u)-2\nu \Theta (\nu \sqrt{u}\left)\right]\\ \times [\nu +\frac{1}{\sqrt{2\pi (t-u)}}\exp (-\frac{{\nu }^2}{2}(t-u\left)\right)-\nu \Theta (\nu \sqrt{t-u}\left)\right]& \text{(2.5.5)}\end{array}$$$$\text{Misplaced \&}$$