= Solution
Apply the multidimensional <Itô formula> to $\xi_t=F(t,S_t,v_t)$. The stated PDE cancels its drift to $rFdt$, leaving
$$
\begin{aligned}
d\xi_t={}&r\xi_tdt
+\sqrt{v_t}\left(S_tF_S+c\rho F_v\right)dW_t\\
&+c\sqrt{v_t}\sqrt{1-\rho^2}F_v\,dZ_t.
\end{aligned}
$$
Consequently $e^{-rt}S_t$ and $e^{-rt}\xi_t$ are local martingales under the physical measure $P$. Thus $P$ itself is an <equivalent local martingale measure> for the augmented market relative to the bank account. The continuous-time <fundamental theorem of asset pricing> says that existence of such a measure for locally bounded prices implies no free lunch with vanishing risk, and hence no arbitrage. The terminal condition also gives $\xi_T=\sqrt{S_T}$ as required.
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