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## COMTE-RENAULT LONG-MEMORY SQUARE ROOT PROCESS

The standard square root process introduced by Feller (1951), used by Cox-Ingersoll-Ross (1985) and used for modeling stochastic volatility by Heston (1993) is the following

$d\tilde{\sigma}^2(t) = k(a-\tilde{\sigma}^2(t)) dt + \gamma \sqrt{\tilde{\sigma}^2} dw(t)$

The Comte-Renault long-memory square root process comes out of two steps.  First, they consider

$dX(t) = -kX(t)dt + \sqrt{X(t) + \tilde{\theta}}dw(t)$ which is equivalent to $\tilde{\sigma}^2(t) = X(t) + \tilde{\theta}$.  Then they consider the long memory process as

$\tilde{\sigma}^2(t) = | \tilde{\theta} + I^{(\alpha)} X(t) |$ where $I^{(\alpha)}$ is the fractional integration operator.  The variance and autocorrelations of $\sigma^(2)$ they relate to the variance and autocorrelations of the square root process $X(t)$ as follows.

(a)  $Var(X(t)) = Var(\tilde{\sigma}^2(t)) = \frac{\tilde{\theta}\gamma^2}{k}$ and

$Var(\sigma^2(t)) = \frac{\theta\gamma^2}{k^{2\alpha+1}} \frac{\Gamma(1-2\alpha)\Gamma(2\alpha)}{\Gamma(1-\ alpha)\Gamma(\alpha)}$

(b) The autocorrelations are $c_X(h) = \frac{\tilde{\theta}\gamma^2}{2k} e^{-k|h|}$ and

$c_{\sigma^2}(h)/c_{\sigma^2}(0) = 1 - \frac{(kh)^{2\alpha+1}}{2\alpha(2\alpha+1)\Gamma(2\alpha)} + O(h^2)$ for small $h$ and

$c_{\sigma^2}(h)/c_{\sigma^2}(0) = (kh)^{2\alpha-1}/\Gamma(2\alpha)$ for large $h$.

(c)  The spectral density of $\sigma^2(t)$ is

$f_{\sigma^2}(\lambda) = \frac{\tilde{\theta} \gamma^2}{\lambda^{2\alpha}(\lambda^2 + k^2)}$