Approximation of A Jump-Diffusion Process
We present a weak convergence of a discrete time process to a jump-diffusion process as the length of sampling interval, h, goes to zero. There is an example given for the weak convergency with using GARCH (1,1)-M model by Engle and Bollerslev(1986). It is shown that ARCH type models can be used as discrete time approximations of jump-diffusion processes. We use Exponential ARCH with Poisson Jump component as an example for the approximation. Therefore, we may use a discrete time ARCH process as an approximation of a jump-diffusion process in estimation and forecasting. And we may use the jump-diffusion process as an approximation of ARCH process when there is distributional results available for the jump-diffusion limit of the sequence of ARCH processes
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