We can rewrite the equation expressing the dynamics of the volatility as

The process {Δ_t} admits an AR(1) representation of the kind

where λ = −1 + p + q is the second eigenvalue of the transition matrix of the Markov chain (the other one being equal to 1) and v_t has the following probability structure (Hamilton 1989, p. 360):

Rearranging equation (1) yields

where e_t = ε_t² − σ_t² is a mean-zero error. Applying the operator (1 − λL) to both sides of (2) and using the AR(1) structure of {Δ_t} yields

The error term on the right-hand side of (3) admits a MA(1) representation

where β is the value less than one in absolute value that, together with σ_u², solves

In the case of interest

with E(Δ_t) = (1 − q)/(2 − p − q).

Therefore we can write (3) as an ARMA(1,1)

where ω = μ₁(1 − λ) + (μ₂ − μ₁)(1 − q), α = −1 + p + q, and β is the solution to (4).

In the special case where p + q = 1 we have that λ = α = β = 0 and ε_t² is a white noise with mean ω = μ₁ + (μ₂ − μ₁)(1 − q).