Finding the maximum likelihood estimator given a special variance

Assuming that $$x_t\sim N(0,\sigma_t^2)$$, where $$\sigma_t^2=\mu+\beta\sigma_t^2+\alpha x_{t-1}^2$$

In this case, what is the likelihood function given the sample data $$(x_1,x_2,\ldots,x_T)$$?

I understand how to solve the typical MLE for a normal distribution by referring to this https://www.statlect.com/fundamentals-of-statistics/normal-distribution-maximum-likelihood but I cannot seem to get the idea on how to solve one with a $$\sigma_t^2$$ related to $$x_{t-1}^2$$.

Any help would be appreciated. Thank you.

• Here is a MathJax tutorial for typesetting. – StubbornAtom Jul 2 at 7:32
• It looks like there must be a typo at the outset, because you attempt to define $\sigma_t^2$ in terms of itself. Please confirm or fix that. – whuber Jul 2 at 13:15