$$dX = \kappa_x (\theta_x - X)dt + \sigma_x \sqrt{X} \,dW_x$$ $$dY = \kappa_y (\theta_y - Y)dt + \sigma_y \sqrt{Y} \,dW_y$$ $$dW_x dW_y = \rho\, dt$$

we know that $X$ and $Y$ are marginally distributed as non-central $\chi^2$. What is the joint distribution of $X$ and $Y$ ?