Let $a, b \in \mathbb{R}$. Show that the stochastic differential equation
$$
d X_{t}=\frac{b-X_{t}}{1-t} d t+d W_{t}
$$
with initial value $X_{0}=a$ has a unique strong solution for $t \in[0,1)$ and that $X_{1}:=$ $\lim _{t \uparrow 1} X_{1}=b$ almost surely. Furthermore, show that the process $Y=\left(X_{t}-a-\right.$ $t(b-a))_{t \in[0,1]}$ can be described by the Itô integral
$$
Y_{t}=(1-t) \int_{0}^{t}(1-s)^{-1} d W_{s}, \quad t \in[0,1)
$$
and is hence a Brownian bridge (compare Exercise 21.5.3).