Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 211 / 3 / a

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- Solution 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

gives

lo g E [e^{θ X_{t}} ∣ X_{t - 1} = x] = a θ x + b θ + ψ (θ) .

Hence the autoregressive process of order one is an affine process, with

A (θ) = a θ, B (θ) = b θ + ψ (θ) .

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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