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

Consider a linear-Gaussian state-observation model on $[0,T]$, with all notation and fixed versions as there.

1. (Regularity and span structure) Each component family $(u^{j}_t)_{t\in[0,T]}$ is mean-square continuous and $u^{j}_0=0$ almost surely. Moreover, for every $t\in[0,T]$, every $X^{i}_t$ and every $u^{j}_t$ is a mean-square limit of finite linear combinations of the random variables $\xi^{i'}$ ($1\le i'\le l$) and $W^{j'}_r$ ($1\le j'\le m$, $0\le r\le t$).

2. (Joint Gaussianity) The combined family consisting of all $\xi^{i}$, all $W^{j}_t$ ($t\ge0$), all $X^{i}_t$, and all $u^{j}_t$ ($t\in[0,T]$), indexed by the disjoint union, is jointly Gaussian.