I'm working on a simple stock pricing model described by the following model of Ito SDE's: $$ d S_t = \mu S_t dt + \sigma_t S_t dB^1_t \\ d \sigma_t = -(\sigma_t - \xi_t)dt + p \sigma_t dB^2 _t \\ d \xi_t = \frac{1}{\alpha}(\sigma_t - \xi_t) dt $$ With $\mu=0.10$, $S_0 = $ €50, $\sigma_0 = 0.20$, $\xi_0 = 0.20$, $\alpha = 1$, and $p=1$.
So I implemented Euler and Milstein schemes for this model, but now I want to study the convergence behaviour of both schemes. In order to do that, I need to know the error to the exact solution, so I would need the exact solution of the model. Using Ito's lemma with $\phi = \ln(S_t)$ gives the following solution for $S_t$, since $\sigma_t$ is independent from $S_t$: $$ S(t) = S_0 e^{(\mu -\sigma_t^2/2)t + \sigma_t B^1(t)} $$
However, I'm really stuck trying to get the exact solution of $\sigma_t$ and $\xi_t$. Could someone please help me out? Any help would be appreciated!