Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 211 / 3 / a / 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 $X_{t-1}=x$ givesHence the autoregressive process of order one is an affine process, withHere 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.