Do we need Feller condition if volatility process jumps?

Question

It is fairly known that in affine processes, as Heston model \begin{equation} \begin{aligned} dS_t &= \mu S_t dt + \sqrt{v_t} S_t dW^{S}_{t} \\ dv_t &= k(\theta - v_t) dt + \xi \sqrt{v_t} dW^{v}_{t} \end{aligned} \end{equation} the SV $v_t$ is a strictly positive process if the drift is stronger enough, i.e. if drift parameters ($k$, the speed of mean-reverting, and $\theta$, mean-reverting level) and the Vol-of-Vol $\xi$ satisfy: \begin{equation} k \theta > \frac{1}{2} \xi^2 \end{equation} which is known as Feller condition. I know this condition can be generalized to multi-factor affine processes. For example, if the volatility of the returns $\log S_t$ is made of several independent factors $v_{1,t},v_{2,t},...,v_{n,t}$, then the Feller condition applies to each factor separately (check here at page 705, for example). Moreover Duffie and Kan (1996) provide a multidimensional extension of the Feller condition.

But I still don't understand if we still need the (or a sort of) Feller condition in case of jump-diffusion. You may consider for example the simple case of a volatility factor with exponentially distributed jumps: \begin{equation} dv_t = k(\theta - v_t) dt + \xi \sqrt{v_t} dW^{v}_{t} + dJ^{v}_{t} \end{equation} where $J^{v}_{t}$ is a compound Poisson process, independent of the Wiener $W^{v}_{t}$. The Poisson arrival intensity is a constant $\lambda$ with mean $\gamma$. I observe that in this case, the long term mean reverting level is jump-adjusted: \begin{equation} \theta \Longrightarrow \theta ^{*}=\theta + \frac{\lambda}{k} \gamma \end{equation} so I suspect if a sort of Feller condition applies it must depends on jumps.

Nevertheless, from a purely intuitive perspective, even if the barrier at $v_t = 0$ is absorbent, jump would pull back from 0 again.

Thanks for your time and attention.

Answer 1

The Feller condition applies without modification.

That is under the assumption that $v$ is square-root process with poisson-arrival jumps (as you wrote), and assuming the jump distribution is strictly positive and initial level $v_0>0$.

The reason is, conditional on no jumps occuring, the process is just a square root process, for which the references you cite show that $v$ remains strictly positive iff $\kappa\theta>\frac{1}{2}\xi^2$. But when a jump arrives, $v$ changes level and resumes following the same process. If the Feller condition is satisfied, it can never diffuse to 0; if the jump distribution is positive it can never jump to 0. For finite arrival rate jumps that covers it: it cannot reach 0. If the Feller condition is not satisfied, since the diffusion is independent of the jump and the no jump probability is positive, even conditional on no jumps the process can reach 0. So again the condition is if-and-only-if.