Proof of Gaussian and Span Structure of the Linear-Gaussian State-Observation Model

Claim 1. The family $\bigl((\tilde E(r)X_r)^{j}\bigr)_{r\in[0,T]}=\bigl(\sum_i\tilde E_{ji}(r)X^{i}_r\bigr)_r$ is mean-square continuous by claims 1-2 of Basic Properties of the Mean-Square Riemann Integral, so its indefinite mean-square Riemann integral is mean-square continuous by claim 6 there; the Wiener parts $\int_0^t\tilde\varepsilon_{jj'}\,dW^{j'}$ are mean-square continuous by part 1 of Integration by Parts for Wiener Integrals and Mean-Square Riemann Integrals; sums are handled by claim 1 of Basic Properties of the Mean-Square Riemann Integral. At $t=0$ all terms vanish by the degenerate-interval conventions recorded in Linear-Gaussian State-Observation Model (via Mean-Square Riemann Integral of a Family of Random Variables and Ito Integrable Process and the Ito Integral), so $u^{j}_0=0$ almost surely.

Span structure. Write $\mathcal{S}_t$ for the closed mean-square span of the collection $\{\xi^{i'}\}\cup\{W^{j'}_r:r\le t\}$; by claim 1 of The Closed Mean-Square Span of a Family of Random Variables it is closed under finite linear combinations and mean-square limits. By claim 3 of Existence, Uniqueness, and Variation of Constants for Linear Stochastic Differential Equations (with vanishing forcing), $X^{i}_s\in\mathcal{S}_t$ for every $s\le t$ (the generating collection for time $s$ is contained in that for time $t$). For $u^{j}_t$: the mean-square Riemann integral $\int_0^t(\tilde EX)^{j}\,dr$ is a mean-square limit of its Riemann sums (Mean-Square Riemann Integral of a Family of Random Variables), which are finite linear combinations of values $X^{i}_\tau$ with $\tau\le t$, all in $\mathcal{S}_t$, so the integral lies in $\mathcal{S}_t$ by the closure of claim 1 of The Closed Mean-Square Span of a Family of Random Variables; the Wiener parts lie in $\mathcal{S}_t$ by the elementary-integral approximation (claim 1 of Adapted Mean-Square Continuous Processes are Ito Integrable with claim 1 of Existence and Uniqueness of the Mean-Square Extension of the Elementary Stochastic Integral: elementary stochastic integrals of the grid-value approximants are finite linear combinations of increments of $W^{j'}$ at times $\le t$). Hence $u^{j}_t\in\mathcal{S}_t$.

Claim 2. Every member of the combined family is either a member of the base family $\mathcal{B}=\{\xi^{i}\}\cup\{W^{j}_t:t\ge0\}$ (jointly Gaussian by the model hypothesis of Linear-Gaussian State-Observation Model) or, by claim 1, a mean-square limit of finite linear — hence affine — combinations of members of $\mathcal{B}$. By Mean-Square Limits of Affine Combinations Adjoin to a Jointly Gaussian Family, the combined family is jointly Gaussian. $\blacksquare$