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## THE PROBLEM OF AUTOCORRELATION OF A FRACTIONAL SQUARE ROOT PROCESS

$D^\mu x_t = -k x_t + \gamma \sqrt{x_t + \theta} dw_t/dt$ (*)

where the second term is a white noise whose expectation is zero

$D_t E(x_{t+h} x_t) = -k E( x_t D^{1-\mu} x_{t+h} + x_{t+h} D^{1-\mu} x_t)$

using $D^1_t x_t = D^\mu_t D^{1-\mu}_t$.   So we get

$D^\mu_t E(x_{t+h} x_t) = -k E( I^{1-\mu} (x_t D_t^{1-\mu} x_{t+h} + x_{t+h}D_t^{1-\mu} x_t) )$ (**)

Now use the formula

$D_t^{1-\mu} (uv) = \sum_{k=0}^\infty Choose(\mu,k) D^{1-\mu-k}_t u D^k_t v$

to see that the right hand side of (**) is

$-k E( x_t x_{t+h} + x_t D^{1-\mu}_t x_{t+h} - \sum_{k=1}^\infty Choose(\mu,k) D^{1-\mu}_t x_t D^k x_{t+h})$

So if second term and the infinite sum have sufficient cancellation, we have

$D^\mu E(x_t x_{t+h}) = -k E( x_t x_{t+h} )$

and then we have the Mittag-Leffler function as the autocorrelation of these fractional square root processes (this is important because we can see that they do match data quite well.)

A direct integration by parts using (*) shows

$D^\mu E(x_t x_{t+h}) = - k E(x_t x_{t+h}) - \int_0^t k^2 E(x_s x_{s+h}) + \gamma^2 \sqrt{(x_s+\theta)(x_{s+h} + \theta)} D^{1-\mu} x_{s+h} ds$

Assuming $0 \le D^{1-\mu} x_{s+h}$ we have the inequality

$D^\mu E(x_t x_{t_h}) + k E(x_t x_{t+h}) \ge 0$ (***)

since the integral is positive.  Consider $g(t), f(t)$ satisfying $f(0) = g(0) = E(x_{0}x_{0+h})$ and

$D^\mu g(t) + k g(t) = 0$(****)

and let $f(t) = E(x_tx_{t+h})$.  Then subtracting (****) from (***) tells us that $D^\mu (f-g)(0) \ge 0$ to which we can apply $D^{1-\mu}$ to ensure that $f(t) \ge g(t)$.  Therefore we have:

The covariance of the process (*) majorizes a Mittag-Leffler function with parameters $E_{\mu}( -kt^\mu )$.  This result is a partial answer to the question for volatility processes since the Mittag-Leffler function has, when $\mu < 1$ slower than exponential decay and therefore the model provides ‘at least long memory’.