Two ARCH models and their limitations as diffusion processes

Two typical ARCH models: the ASDARCH model and the APARCH model are analyzed. Let Y
k
, and σ
²
k
denote the log returns and the volatility. When the time interval h goes to zero, (Y
k
; σ
²
k
), as a discrete time Markov chain system, weakly converges to a continuous time diffusion process. The continuous time approximation of the ASDARCH model is done using two different methods. With some transformation, these two results are equivalent to high frequency data. The continuous time approximation of the APARCH model is obtained by a different procedure.