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An autoregressive process of order one with independent and identically distributed random variables as innovations is an affine process when the innovation exponential moments are finite. Its one-step coefficients are $A(\theta )=a\theta$ and $B(\theta )=b\theta +\psi (\theta )$, where $\psi$ is the innovation cumulant-generating function.

If a discrete-time martingale deflator obeys $Y_t/Y_{t-1}=e^{X_t}$ for an affine process $X$, unit zero-coupon bonds have exponential-affine prices. For one-step coefficients $A,B$, the recursion is $\alpha (0)=\beta (0)=0$, $\alpha (n+1)=A(1+\alpha (n))$, $\beta (n+1)=\beta (n)+B(1+\alpha (n))$.

For a scalar discrete-time affine process, the conditional cumulant-generating function of ${\sum }_{j=1}^t{\theta }_j{X}_j$ is affine in $X_0$. Backward effective parameters $q_t={\theta }_t$, $q_j={\theta }_j+A(q_{j+1})$ give coefficients $A(q_1)$ and ${\sum }_{j}B(q_j)$.

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.

Use the time-homogeneous Markov property and define backward effective parameters byConditioning the last exponential factor on ${\mathcal{F}}_{t-1}$ replaces ${\theta }_t{X}_t$ by $A(q_t)X_{t-1}+B(q_t)$. Repeating the tower property of conditional expectation combines this with the preceding exponent, then with each earlier exponent. The last remaining conditional transform is at time $1$, soThese are finite because the one-step affine process transforms are finite at every real parameter. The argument establishes the entire joint affine transform, including when the coefficients ${\theta }_j$ have different signs.

A unit-face-value zero-coupon bond pays $1$ at maturity. The martingale deflator pricing identity isUse the affine process Markov property with respect to the market filtration; if that filtration contains extra predictive information, the natural Markov property alone would not suffice. Part (b), with every future coefficient equal to $1$, gives the exponential-affine form. More explicitly, the exponential-affine bond pricing recursion isIndeed, conditioning the first future step in an $(n+1)$-step horizon transforms $e^{(1+\alpha (n))X_{t+1}+\beta (n)}$ into $e^{A(1+\alpha (n))X_t+B(1+\alpha (n))+\beta (n)}$. ThereforeThis uses the true martingale deflator pricing identity; a merely local martingale deflator would not by itself justify replacing prices by conditional terminal expectations.