= Solution
Assume the innovations are <independent> of the previous history and have finite <exponential moments> at every real argument, as required by the finite-valued affine definition. Conditioning on $X_{t-1}=x$ gives
$$
\log\mathbb E[e^{\theta X_t}\mid X_{t-1}=x]=a\theta x+b\theta+\psi(\theta).
$$
Hence the <autoregressive process of order one> is an <affine process>, with
$$
\boxed{A(\theta)=a\theta,\qquad B(\theta)=b\theta+\psi(\theta).}
$$
Here and in the remaining parts, an <affine process> uses a time-homogeneous <Markov process>: the one-step transform is the same at every time. The displayed definition at time $1$ alone would not determine later transitions of an arbitrary time-inhomogeneous <Markov process>.
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