I am reading chapter 24 of Kallenberg's modern probability. In the proof of theorem 24.1
Cauchy Problem: $CP(f,v):\dot{u}=Au-vu$ and $u(0,x)=f(x)$.
Theorem 24.1. (Cauchy problem, Feynman, Kac) Let $A$ with domain $D$ be the generator of a Feller diffusion $X$ in $\mathbb{R}^d$, and fix $u\in C_b(\mathbb{R}^d)$ and $v\in C_b^+(\mathbb{R}^d)$. Then any bounded solution $u:[0;\infty)\times\mathbb{R}^d\rightarrow\mathbb{R}$ to $CP(f,v)$, that is $C^1$ in the time-variable and $C^2$ in the space variable, is given by $$ u(t,x)=\mathbb{E}_x[e^{-V_t}f(X_t)], $$ where $V_t=\int_0^t v(X_s)ds$. Conversely the function $u$ above solves $CP(f,v)$ if $f\in D$.
they define $M_s=e^{-V_s}u(t-s,X_s)$ for fixed $t$ and $s\in[0;t]$ and show that it is a local martingale. However they use Îto's formula, i.e. they try to express $du(t-s,X_s)$ in terms of a second order expansion. But I don't understand why we are allowed to use Îto's formula, since we didn't show that $X_s$ is a semimartingale. So why is $X_s$ is a semimartingale?
They mention Dynkin's formula, Lemma 19.21,
Lemma 19.21 (Dynkin's formula). The processes $M^h_t=h(X_t)-h(X_0)-\int_0^t Ah(X_s)ds$ for $h\in D$ are martingales.
so do they say that by choosing a suitable $h$ we can see that $X_t$ is a semimartingale? We know that $C_c^\infty(\mathbb{R}^d)\subseteq D$ by definition of Feller diffusion. So we can try to plug in some $h$ that look close to $X_t$ like the identity and elsewhere are $0$. Since $A$ is local, $Ah$ will vanish there, so $M^h_t$ will look like $X_t$.
Edit: I also have difficulties in the proof of the above theorem to understand how Itôs formula is applied. They write $$ dM_s = e^{-V_s}(du(t-s,X_s)-u(t-s,X_s)v(X_s)ds) $$ by integration by parts. But then they mention Dynkin's formula and Itô's formula to write further $$ dM_s = e^{-V_s}(Au(t-s,X_s)-\dot{u}(t-s,X_s)-u(t-s,X_s)v(X_s))ds + L_s, $$ where $L_s$ is some local martingale. I don't understand at all, how $A$ comes into play when using Itô's formula on $u(t-s,X_s)$. When applying Itô's formula, we get in Einstein summation $$ du(t-s,X_s)=\partial_ju(t-s,X_s)dX_s^j + \partial_i\partial_ju(t-s,X_s)d[X^i,X^j]_s. $$ The local martingale part from the $dX_s^j$-integration can be absorbed into $L_s$, but how to relate all the other terms to $A$? What is $[X^i,X^j]$ in this case? I know that there is a formula relating a second order expansion with the generator of a Feller diffusion, Theorem 19.24 in Kallenberg, but nothing is said about uniqueness.
You're on the right track. For each $x\in\Bbb R^d$ and $r>0$ let $f$ be a smooth continuous function of compact support such that $f(y) = y$ for all $y\in B(x,r)$. Then $f$ is in $D$ and so $f(X)$ is a semimartingale under $\Bbb P_x$. Therefore $X$ is a semimartingale up until the exit time $\inf\{t: X_t\notin B(x,r)\}$. As $r$ can be made as large as you please, this shows that $X$ is a local semimartingale, hence a semimartingale under $\Bbb P_x$.