Let $(v_t)$ be a discrete time series of variance obeying a mean-reverting variance process $v_t$, which is actually the discrete version of the Heston model in finance. \begin{align} x_t &= \sqrt{v_t} z_{1,t} \tag1\\ v_{t}-v_{t-1} &= -\lambda(v_{t-1}-v_\infty)+\eta\sqrt[]{v_{t-1}}z_{2,t-1} \tag2 \end{align} where $z_{1,t}$ and $z_{2,t}$ is two not necessarily independent unit variance random walks with correlation and $\eta$ is some positive constant.

I would like to make an approximate GARCH(1,1) model for the variance out of the above time series in the form of $$v_t = \alpha_0+\alpha x_{t-1}^2+\beta v_{t-1}$$ where $\alpha_0,\,\alpha,\,\beta$ are positive and $\alpha+\beta<1$.

Here is my rough ad hoc attempt. Take expectation of Equation (2) and arrange the terms we have $$u_t = \lambda u_\infty+(1-\lambda)u_{t-1}$$ where $u_t:=\mathbf E[v_t]$. Now set $$\tilde u_t = \lambda \tilde u_\infty+(1-\lambda)(ax_{t-1}^2+b\tilde u_{t-1})$$ for some positive $a$ and $b$ where $a+b=1$. Now $\alpha_0 = \lambda u_\infty,\,\alpha=(1-\lambda)a,\,\beta=(1-\lambda)b$. How would one determine $a,\,b$ from the time series recursion Equation (1) and (2)?


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