This file implements the time integration of a system of stochastic differential equations (SDEs), with linear drift and constant diffusion, whose invariant is the joint normal distribution.
In a nutshell, the equation integrated governs a set of scalars, , , as
with parameter vectors , , and symmetric positive semi-definite diffusion matrix . Here is an isotropic vector-valued Wiener process with independent increments. The invariant distribution is the joint normal distribution. This system of SDEs consists of N coupled equations, each being a single-variate Ornstein-Uhlenbeck process.
From the Fokker-Planck equation, equivalent to the SDE above, the equations governing the means, , are
while the equation governing the covariance matrix, , is
Ornstein-Uhlenbeck SDE used polymorphically with DiffEq.
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