Can you give me some clue on proofing that squared log-returns of a GARCH(1,1) time series is ARMA(1,1)?

Question

I am working on an exercise from René Carmona's Statistical Analysis of Financial Data in R and I am stuck at this exercise.

The problem

I have a series $\{Y_t\}_t$ of log-returns from an asset that has a $\text{GARCH}(1,1)$ representation of the form:

$$Y_t=\sigma_t\tilde{\epsilon_t}\ \ \ \ \sigma_t^2 = c+ b\sigma_{t-1}^2+aY_{t-1}^2$$

where we assume that $\{\tilde{\epsilon}_t\}_t$ is strong $\text{N}(0,1)$ white noise, and where the coefficients $a,b$ and $c$ are such that $\sigma_t^2$ is stationary.

I want to proof that

$$Y_t^2=c+(b+a)Y_{t-1}^2+\epsilon_t-b\epsilon_{t-1}$$

for some weak white noise $\{\epsilon_t\}_t$ which you should identify.

What I've done?

I read again the parts of the book that talks about GARCH and didn't find any useful theorem. I guess I have to use the regular toolkit of probabilities and stochastic processes.

I have a feeling that I should be able to find a power series somewhere and use the properties of $a,b$ and $c$ to prove that it converges. But I can be completely wrong.

Can you give me some hint?

Answer 1

I found the solution!

The path is to split

$$Y_t = \sigma_t + \sigma_t(\epsilon_t-1)$$

and realizing that $\sigma_t(\epsilon_t - 1)$ is a weak white-noise.