Solution
= Solution
On the finite state space, $P_t=e^{tQ}$ and
$$
\mathbb E_xf(X_t)=(P_tf)(x).
$$
Therefore
$$
\lim_{t\downarrow0}\frac{\mathbb E_xf(X_t)-f(x)}t
=\left(\lim_{t\downarrow0}\frac{P_t-I}{t}f\right)(x)
=Qf(x).
$$
The <matrix> semigroup satisfies $\frac d{dt}P_tf=P_tQf$. Integrating from $0$ to $t$ gives the backward <integral> equation
$$
\boxed{
\mathbb E_xf(X_t)=f(x)+\int_0^t\mathbb E_x[Qf(X_s)]\,ds.}
$$