It is well known that the solution to an ordinary stochastic differential equation has the Markov property so that if one tries to model some kind of long memory process one has to instead consider SDE's driven by e.g. fractional brownian motions.
I've recently heard mentioned that one can also get this behaviour by superimposing SDE's (i.e. just adding them together, right?) intuitively how come this is possible? Does anyone have a readable source to this phenomenon.
Thanks,
Take $A=\text{diag}(a_1,...,a_n)$, $a_i>0$ and consider the vector SDE ($n$ components) \begin{equation} dX_t=-A X_t +\sigma dW_t, \end{equation} where $W_t$ is a vector of independent Wiener processes and $\sigma>0$ a scalar. This is a special case of the vector OU (Ornstein-Uhlenbeck) process. Its temporal autocovariance reads \begin{equation} \Bigl\langle X_tX_{t+\tau}^{\top} \Bigr\rangle=\text{diag}\left(\frac{\sigma^2}{2a_1} e^{-a_1 |\tau|},...,\frac{\sigma^2}{2a_n} e^{-a_n |\tau|}\right), \end{equation} which means that the components of $X_t$ are independent. Thus, if you take a linear combination \begin{equation} y_t=w^{\top}X_t, \end{equation} it will have autocovariance \begin{equation} \Bigl\langle y_ty_{t+\tau} \Bigr\rangle=\sum_{i=1}^n\frac{w_i^2\sigma^2}{2a_i} e^{-a_i |\tau|}, \end{equation} which allows you to mix $a_i$'s (inverse time-constants) of different scales. Although the tail of the autocovariance function will still decay exponentially, for processes with finite time horizon $T$ this is not a big issue, since by incorporating a time-constant on the order of $T$ will let the process have long memory. Moreover, you can approximate a power law quite well with a large number of exponentials.