Mathematica code for numerical generation of random process with given distribution and exponential autocorrelation function

Stochastic simulations commonly require random process generation with a predefined probability density function (PDF) and an exponential autocorrelation function (ACF). Such processes may be represented as a solution of a stochastic differential equation (SDE) of the first order. The numerically-stable solution of this SDE may be provided by a discrete-time differential equation. Both the generation of the required SDE and the implementation of the differential equation may be effectively done by Mathematica software for most of the typical distributions. Moreover, the required implicit Milstein method for positive domain distributions is not supplied by built-in SDE-related Mathematica functions.