 Solution

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.