I have troubles understanding part (b) of the proof to Theorem 8.1.1 in the Øksendal textbook (https://link.springer.com/book/10.1007/978-3-642-14394-6)
The theorem is as follows:
Let $X_t$ be an Itô diffusion in $\mathbb{R}^n$ with generator $A$ and $f\in C_0^2(\mathbb{R}^n)$.
(a):
Define \begin{equation*} u(t,x)=E^x[f(X_t)]. \end{equation*} Then $u(t,\cdot)\in\mathcal{D}_A$ for each $t$ and \begin{align} \frac{\partial u}{\partial t}=Au,~ t>0,~x\in\mathbb{R}^n\\ u(0,x)=f(x),~x\in\mathbb{R}^n \end{align} where the right hand side is to be interpreted as $A$ applied to the function $x\mapsto u(t,x)$.
(b):
Moreover, if $w(t,x)\in C^{1,2}(\mathbb{R}\times\mathbb{R}^n)$ is a bounded function satisfying \begin{align} \frac{\partial w}{\partial t}=Aw,~ t>0,~x\in\mathbb{R}^n\\ w(0,x)=f(x),~x\in\mathbb{R}^n \end{align} then $w(t,x)=u(t,x)$.
The proof of part (b) starts as follows:
Fix $(s,x)\in\mathbb{R}\times\mathbb{R}^n$. Define the process $Y_t$ in $\mathbb{R}^{n+1}$ by $Y_t=(s-t,X_t^x)$, $t\geq 0$. Then $Y_t$ has generator $\tilde{A}$ with \begin{equation*} \tilde{A}w(t,x)=-\frac{\partial}{\partial t}w(t,x)+Aw(t,x)=0,~t>0,~x\in\mathbb{R}^n. \end{equation*} By Dynkin's formula we have for all $t\geq 0$ \begin{equation*} E^{(s,x)}[w(Y_{t\wedge\tau_R)})]=w(s,x)+E^{(s,x)}\left[\int_0^{t\wedge\tau_R}\tilde{A}w(Y_r)dr\right], \end{equation*} where $\tau_R:=\inf\{t>0||X_t|\geq R\}$.
Now we want to conclude that \begin{equation*} E^{(s,x)}[w(Y_{t\wedge\tau_R)})]=w(s,x). \end{equation*} It seems as Øksendal uses the fact that $\tilde{A}$ vanishes, but this is only the case for $t>0$, so the first component of $Y_r$ has to be positiv? But why is this the case? Or what do I misunderstand?
In the integral $0<r<t\wedge\tau_R\le t$, so the first component of $Y_r$, namely $s-r$, is in the interval $(s-t,s)$. You are right to be wary: Øksendal's deduction of the second equality in the formula appearing in the first display following (8.1.6) requires $t$ to be at most $s$. But this restriction is okay, because the identity is subsequently used only for $t=s$.