This paper extends exit theorems of Da Prato and Zabczyk to nonconstant diffusion coefficients.It uses extensively general, exponential estimates due to Peszat.
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.
