I am currently trying to transform a discrete time stochastic model into a continuous time SDE. The model I have been given contains "red noise" as its driving force: $$X_{k+1}=(1-\lambda) X_k + U_k$$ $$U_{k+1}=\alpha U_k+\epsilon_k$$ where $0<\alpha<1,0<\lambda<1$, the $\epsilon_k$ are independent gaussian random variables (simulating discrete time white noise) and $U$ is therefore a discrete realization of an Ornstein Uhlenbeck process , i.e. an AR(1) process (let me know whether you agree with this last statement). Now, I have seen in several places in the literature that the corresponding "SDE" should be $$dX_t=-\lambda X_tdt+U_tdt$$ $$dU_t=-\theta U_tdt+\sigma dW_t$$ where you choose $\theta$ corresponding to $\alpha$ by $\alpha=e^{-\theta}$. I see where they might be coming from, since discretizing this at the right sampling rate will give you back the original model. However, if you sample at a finer rate, you will see very different dynamics. $X_t$ will be differentiable in this model and the equation above is not really an SDE. Most people in stochastics will not consider $U_tdt$ as noise.
Instead, some people propose this SDE $$dX_t=-\lambda X_tdt+dU_t$$ $$dU_t=-\theta U_tdt+\sigma dW_t$$ I would say that this model also does not give you what you want. When you discretize this back, the increments $\Delta U_t=U_{t+\Delta t}-U_t$ do not have the spectrum you would want (in the red noise case, you want a spectrum $\sim \omega^{-2}$). Instead the increments have more of a "blue noise" spectrum, i.e. high power density in high frequencies instead of low. The increments are also negatively correlated, which means they can not be used to model persistence in time.
My big question is: Have you ever seen a continuous time stochastic model that is supposed to be driven by "red noise"? Can it even be written as an SDE? My gut feeling says no, because whenever you introduce the term $dW_t$, you will get a constant summand of 1 to your spectrum.
I hope it is somewhat understandable what I am looking for, let me know if you would like further elaboration on the relation of AR(1) $\leftrightarrow$ OU-process in the first part.
Thanks!
