Is Heston model an affine jump-diffusion?

Question

In Duffie, Pan and Singleton's paper "Transform Analysis and Asset Pricing for Affine Jump-diffusions" (2000) they define affine jump-diffusion (AJD) a process of the following form: $$dX_t=\mu(X_t)dt+\sigma(X_t)dW_t+dZ_t,$$ where $\mu$, $\sigma$ are affine functions, $W$ is a (multivariate) Wiener process and $Z$ is a compound Poisson Process.
In the same paper they say "...there is a substantial literature building on the particular affine stochastic-volatility model for currency and equity prices proposed by Heston (1993)"; however, Heston model has the following form: \begin{equation} \begin{cases} d{S_t}=\mu{S_t}dt+\sqrt{v_t}S_t{d{W_t^1}}\\ d{v_t}=k(\theta-v_t)dt+\sigma\sqrt{v_t}{d{W_t^2}}\\ d{\bigl\langle{W^1,W^2}\bigl\rangle}_t=\rho{dt}. \end{cases} \end{equation} Now, how can they affirm that this is an affine model? The volatility $v_t$ depends on his square root and if it was affine than the volatility should depend on $v_t$. What am I missing?

Answer 1

In the stochastic volatility literature, a diffusion model for $X_t$ is called "affine" if it has a following conditional characteristic function of the following form:

$$\mathbb{E}\left( uX_t \, | \, \mathcal{F_t} \right) = \exp \left( \phi (T-t) + \psi (T-t) X_t \right) $$

In particular, the conditional characteristic function is the exponential of an affine transformation of the process $X_t$. The standard Heston model does indeed admit a characteristic function of this form where $X_t$ is the log-price process.