# Mean and Variance in the Jacobi Stochastic Volatility model

I would like to compute $$E[X_{T}]$$ and $$Var[X_{T}]$$ in the Jacobi model, where the Dynamics are given as \begin{align} dY_{t}&=\kappa(\theta-Y_{t})dt+\sigma\sqrt{Q(Y_{t})}dW_{1t}\\ dX_{t}&=(r-\delta-0.5Y_{t})dt+\rho\sqrt{Q(Y_{t})}dW_{1t}+\sqrt{Y_{t}-\rho^{2}Q(Y_{t})}dW_{2t} \end{align} where $$W_{1t}$$ and $$W_{2t}$$ are two correlated Brownian motions and $$Q(Y_{t})=\frac{(Y_{t}-y_{min})(y_{max}-Y_{t})}{(\sqrt{y_{max}}-\sqrt{y_{min}})^2}$$. The parameters are given as: $$\kappa=0.5=-\rho, r=\delta=X_{0}=0, \theta=Y_{0}=0.04, y_{min}=10^{-4}, y_{max}=0.08$$ and $$\sigma=1$$ and $$T=1/12$$.