 Solution

Apply the multidimensional Itô formula to ${\xi }_t=F(t,S_t,v_t)$. The stated PDE cancels its drift to $rFdt$, leavingConsequently $e^{-rt}{S}_t$ 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.