Prove that the process [X_{t}=exp left(a B_{t}+b t ight)left(x+int_{0}^{t} d s exp left(-a B_{s}-b s ight) ight)]

Prove that the process

\[X_{t}=\exp \left(a B_{t}+b t\right)\left(x+\int_{0}^{t} d s \exp \left(-a B_{s}-b s\right)\right)\]

satisfies

\[X_{t}=x+a \int_{0}^{t} X_{u} d B_{u}+\int_{0}^{t}\left(\left(\frac{a^{2}}{2}+b\right) X_{u}+1\right) d u\]

More generally, consider the process

\[d Y_{t}=\left(a Y_{t}+b\right) d t+\left(c Y_{t}+d\right) d W_{t},\]

where \(c eq 0\). Prove that, if \(X_{t}=c Y_{t}+d\), then

\[d X_{t}=\left(\alpha X_{t}+\beta\right) d t+X_{t} d W_{t}\]
with \(\alpha=a /c, \beta=b-d a / c\). From \(T_{\alpha}\left(Y^{y}\right)=T_{c \alpha+d}\left(X^{c x+d}\right)\), deduce the Laplace transform of first hitting times for the process \(Y\).