I have comme across a paper in my research, which attempts to model a stochastic process $X_t = \bar{X_t} + \Delta X_t$ with a known mean part $\bar{X_t}$ and a fluctuating error around the mean $\Delta X_t$.
The paper gives details of how the fluctuating error term is distributed : $$ p\left(\Delta X_t \vert t\right) = \frac{\left|\frac{\Gamma\left(m+\frac{\nu}{2} i\right)}{\Gamma(m)}\right|^{2}}{\alpha \mathrm{B}\left(m-\frac{1}{2}, \frac{1}{2}\right)}\left[1+\left(\frac{x-\lambda}{\alpha}\right)^{2}\right]^{-m} \exp \left[-\nu \arctan \left(\frac{x-\lambda}{\alpha}\right)\right] $$
Where $m, \nu, \alpha, \lambda$ are parameters of the Pearson Type 4 distribution dependent on t, and $B, \Gamma$ are repectively the Beta and Gamma Functions.
The objective of the paper was to derive this distribution and to link it's parameters to some physical quantities of interest, which is in itself usefull information.
However, I was wondering if it was possible to go beyound and to generate realisation of the compelte stochastic process $X_t$ for which the mean $\bar{X_t}$ is known.
The first idea that commes to mind is to add randomly generated samples drawn from $p\left(\Delta X_t \vert t\right)$ to the mean at each timestep $t$. However, I am concerned that this might give a rather discontinuous result, which would be somewhat in opposition with the underlying physics.
My question is therefore if there is a way to generate a smoooth realisation of this stochastic process.
I have started to look around for answers, looking at Weinstein, Brownian and O-U processes, but I have failed to find how to fit my problem within these frameworks.