Gaussian Structure and Moment Equations for Linear Stochastic Differential Equations

Throughout, a real-valued function on a subinterval $I$ of the real numbers $\mathbb{R}$ is called continuous on $I$ when it is continuous relative to $I$, both $I$ and the codomain $\mathbb{R}$ carrying the metric of the real line. Let $(A,0,\varepsilon,\xi,W)$ be a linear stochastic differential equation with additive Wiener noise on $[0,T]$ whose forcing family has every member equal to the zero tuple, with dimensions $l,m$, and let $X$ be any mean-square solution of it, with versions fixed (any two mean-square solutions agree almost surely at each time by claim 2 of Existence, Uniqueness, and Variation of Constants for Linear Stochastic Differential Equations, and all conclusions below are unaffected by almost-sure replacement, by Almost Sure Modifications of Gaussian Random Vectors are Gaussian and invariance of moments). Suppose that the combined family of the components $\xi^{i}$ ($1\le i\le l$) and the values $W^{j}_t$ ($1\le j\le m$, $t\ge0$), indexed by the disjoint union, is jointly Gaussian.

1. (Joint Gaussianity) The combined family consisting of all $\xi^{i}$, all $W^{j}_t$, all Wiener integrals $\int_0^{s}f(u)\,dW^{j}_u$ (over all $j$, $s>0$, and continuous $f:[0,s]\to\mathbb{R}$, any versions), and all $X^{i}_t$ ($1\le i\le l$, $t\in[0,T]$), indexed by the disjoint union, is jointly Gaussian.

2. (Mean equation) The functions $\mu^{i}(t)=\mathbb{E}[X^{i}_t]$ are continuous and satisfy, componentwise with the Riemann integral,

where $\mathbb{E}[\xi]$ is the tuple of the $\mathbb{E}[\xi^{i}]$; consequently $\mu(t)=\Phi(t)\,\mathbb{E}[\xi]$ with the fundamental solution $\Phi$ of Fundamental Solution and Variation of Constants for Linear Ordinary Differential Equations.

3. (Covariance equation) Suppose additionally that $\sigma(\xi^{1},\dots,\xi^{l})$ is independent of $\sigma(W^{j}_t:1\le j\le m,\ t\ge0)$, with the generated $\sigma$-algebras. Then the matrix function $P(t)=\bigl(\operatorname{Cov}(X^{i}_t,X^{j}_t)\bigr)_{1\le i,j\le l}$, with the covariance, has continuous entries and satisfies, entrywise,

where $P_0=\bigl(\operatorname{Cov}(\xi^{i},\xi^{j})\bigr)$ and $(\cdot)^{\top}$ is the transpose; hence $P$ is the unique solution of Lyapunov Representation and Positive Semidefiniteness for Linear Matrix Equations for the data $(A,\varepsilon\varepsilon^{\top},P_0)$.