Diffusion-type models with given marginal distribution and autocorrelation function

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Volatility is rough (2014, unknown): “…one of the main messages of our work is that our model is able to replicate the stylized facts of the time series without having to appeal to jumps, see Bibby et al. (2005) for another example where a continuous process can mimic the properties obtained from data generated by a process with jumps.”
The Pearson Diffusions: A Class of Statistically Tractable Diffusion Processes (2008, methodology): “An autocorrelation function of the form (32) fits turbulence data well, see Barndorff-Nielsen et al. (1990) and Bibby et al. (2005).”