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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 ...


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First, I don't see $S^0$ appear anywhere, so I assume it is just used somewhere else. Second, there is probably a point (i) in your question, because you included the point (ii). I'd expect that $\xi_0$ is actually defined there. If that's the case, then $\xi_t$ is predictable. Otherwise, if $\xi_0$ is not defined anywhere then the process $\xi_t$ is ...


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If we are going to have the form \begin{align*} dr = A dt + BdW_t, \end{align*} Then both A and B are functions of $t$ and $r_t$, otherwise, $r_t$ is normal. However, note that \begin{align*} r_t = \exp\Bigg(\frac{1}{\sigma(t)}\bigg(\int_0^t \theta(s)\sigma(s) ds +\sigma(0)\ln r_0 + \int_0^t\sigma^2(s) dW_s\bigg)\Bigg). \end{align*} That is, $r_t$ is ...



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