F. Mainardi (mainardi-2014-approximations-mittag-leffler-function) introduced the following approximation for the Mittag-Leffler function when
with . By Taylor expansion
This is then the behavior of the ‘autocorrelation’ function satisfying . This elementary example provides some intuition for stochastic models where the autocorrelation function can be fit by Mittag-Leffler functions.
The Caputo derivative is more useful for computations because one can inherit the product rule from the usual derivative and constants are annihilated by it.
Suppose . Then the product rule inside the integral and integration by parts gives
We can now add stochastic components $D^\mu x_t = -\lambda x_t + A dw_t$ with zero expectations and still have
where . Assuming this is small, the Mittag-Leffler function will be a good approximation for the autocorrelation function of .