I am struggling with showing the unique solution of SDE and I don't understand the solution at all. Hence I was hoping if someone could make it simpler and explain it to me. Thank you
Show that $$ r(t)=r(0) e^{-\beta t}+\frac{b}{\beta}\left(1-e^{-\beta t}\right)+\sigma e^{-\beta t} \int_{0}^{t} e^{\beta s} d W_{s}^{*} $$ is the unique solution to the Vasiček short-rate equation $$ d r_{t}=\left(b-\beta r_{t}\right) d t+\sigma d W_{t}^{*} $$
Solution
The fact that $r(t)$ is a solution follows from simple calculus. Let us prove that the solution is unique. Indeed if we have two solutions $r, \tilde{r}$ then for $u=\tilde{r}-r$ we get $u(0)=0$ and $u^{\prime}(t)=-\beta u(t)$ and so $u \equiv 0$ as required.
I specifically don't understand how the $u^{\prime}(t)$ was found and why $u=\tilde{r}-r$ implies $u(0)=0$ as both $r, \tilde{r}$ are different solutions.
The two solutions $r$ and $\tilde r$ are meant to have the same initial point, $r_0=\tilde r_0$, thus for their difference you get $u_0=0$.
As the equation is linear in $r$, you can subtract the two instances and get an equation for the difference, \begin{align} d r_{t}&=\left(b-\beta r_{t}\right) d t+\sigma d W_{t}^{*} \\ d \tilde r_{t}&=\left(b-\beta \tilde r_{t}\right) d t+\sigma d W_{t}^{*} \\ \hline d(r_{t}-\tilde r_{t})&=-\beta r_{t}\,d t+\beta \tilde r_{t}\, d t \\ du_t&=-\beta u_t\,dt \end{align}
The general uniqueness statement (along with existence) for SDE also follows from the Lipschitz condition for its coefficients, similar to the ODE result. Linear functions are always Lipschitz (in finite dimensions).