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MPIPKS:
Stochastic processes
Abstract:
We consider autoregressive conditional heteroskedasticity (ARCH) processes in which the variance sigma(y)(2) depends linearly on the absolute value of the random variable y as sigma(y)(2)=a+b\y\. While for the standard model, where sigma(y)(2)=a+by(2), the corresponding probability distribution function (PDF) P(y) decays as a power law for \y\-->infinity, in the linear case it decays exponentially as P(y)similar toexp(-alpha\y\), with alpha=2/b. We extend these results to the more general case sigma(y)(2)=a+b\y\(q), with 0<q<2. We find stretched exponential decay for 1<q<2 and stretched Gaussian behavior for 0<q<1. As an application, we consider the case q=1 as our starting scheme for modeling the PDF of daily (logarithmic) variations in the Dow Jones stock market index. When the history of the ARCH process is taken into account, the resulting PDF becomes a stretched exponential even for q=1, with a stretched exponent beta=2/3, in a much better agreement with the empirical data.
