Wiener Integrals Against a Vector Brownian Motion are Jointly Gaussian

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 $(\Omega,\mathcal{F},P)$ be a probability space, let $m\ge1$ be a natural number, and let $W=(W^{1},\dots,W^{m})$ be an $m$-dimensional Brownian motion on $(\Omega,\mathcal{F},P)$. For each component index $j$, each real $s>0$, and each continuous $f:[0,s]\to\mathbb{R}$, let $\int_0^{s}f(u)\,dW^{j}_u$ denote a Wiener integral of $f$ with respect to the standard Brownian motion $W^{j}$, that is, the It^{o} integral of $(f(u))_{u\in(0,s]}$ with respect to the It^{o} integrator $(W^{j},1)$ of Brownian Motion is an Ito Integrator with Unit Intensity, taken with respect to the natural filtration of $W^{j}$.

1. (Measurable versions) For each $j$, $s$, and $f$ as above, some version of $\int_0^{s}f(u)\,dW^{j}_u$ is measurable with respect to the generated $\sigma$-algebra $\sigma(W^{j}_t:t\ge0)$; indeed one may choose a version measurable with respect to $\sigma(W^{j}_u:0\le u\le s)$.

2. (Joint Gaussianity) For every choice of versions, the family consisting of all the random variables $W^{j}_t$ ($1\le j\le m$, $t\ge0$) together with all the Wiener integrals $\int_0^{s}f(u)\,dW^{j}_u$ (over all $1\le j\le m$, all real $s>0$, and all continuous $f:[0,s]\to\mathbb{R}$), indexed by the disjoint union of the set of pairs $(j,t)$ and the set of triples $(j,s,f)$, is jointly Gaussian, and all its members have expectation zero, the expectations being defined and finite by Square-Integrability, Moments, and Covariance Matrix of a Gaussian Random Vector.

3. (Covariances) With the covariance of square-integrable random variables (defined and finite by Square-Integrability, Moments, and Covariance Matrix of a Gaussian Random Vector), the Riemann integral (existing by claim 3 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval), and the convention that an integral over the degenerate interval $[0,0]$ equals $0$: for all component indices $i,j$, all reals $s,t>0$, and all continuous $f:[0,s]\to\mathbb{R}$ and $g:[0,t]\to\mathbb{R}$,

and, now allowing also $t\ge0$ and $s'\ge0$: $\operatorname{Cov}\bigl(W^{i}_t,\int_0^{s}f(u)\,dW^{j}_u\bigr)$ equals $\int_0^{\min(s,t)}f(u)\,du$ when $i=j$ and equals $0$ when $i\ne j$, and $\operatorname{Cov}(W^{i}_{s'},W^{j}_t)$ equals $\min(s',t)$ when $i=j$ and $0$ when $i\ne j$.