(a) Prove that the Ricciardi and Sato result given in Comments 3.4.1.2.

(b) allows us to express the density of

\[\tau:=\inf \left\{t: x+W_{t}=\sqrt{1+2 k t}\right\}\]

The hitting time of \(a\) for an OU process is

\[\inf \left\{t: e^{-k t}\left(x+\widehat{W}_{A(t)}\right)=a\right\}=\inf \left\{u: x+\widehat{W}_{u}=a e^{k A^{-1}(u)}\right\} .\]

Comments 3.4.1.2:

\[-k e^{k\left(x^{2}-a^{2}\right) / 2} \sum_{n=1}^{\infty} \frac{D_{u_{n, a}}(x \sqrt{2 k})}{D_{u_{n, a}}^{\prime}(a \sqrt{2 k})} e^{-k u_{n, a} t}\]