Consider the two-dimensional SDE \begin{align} dS_t &= \mu S_t dt + S_t \sqrt{V_t} \big(\rho dW_t^{(1)}+\sqrt{1-\rho^2}dW_t^{(2)} \big); \tag*{(1)} \\ dV_t &= \kappa(\theta - V_t)dt + \sigma \sqrt{V_t}dW_t^{(1)}, \tag*{(2)} \end{align} where $(W^{(1)},W^{(2)})$ is a two-dimensional standard Brownian motion and $\mu, \rho, \kappa, \theta$ and $\sigma$ are constants. We assume that the Feller condition is satisfied, i.e. $$2 \kappa \theta > \sigma^2,$$ which ensures that $V_t >0.$
I have read in a book that the solution $(S_t,V_t)_{0 \leq t \leq T}$ to the two-dimensional SDE above is a Markov process but they don't give any proof. I've been looking for sufficient conditions for the solution to be a Markov process. The only condition I found requires the coefficients (drift and diffusion functions) to be Lipschitz and have linear growth. But this is not the case for this SDE, so I don't know how to proceed. Any ideas?