Wiener Integrals of Continuous Functions 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 $B=(B_t)_{t\ge0}$ be a standard Brownian motion on a probability space $(\Omega,\mathcal{F},P)$, with its natural filtration $(\mathcal{F}^{B}_t)_{t\ge0}$, regarded as the It^{o} integrator $(B,1)$ of Brownian Motion is an Ito Integrator with Unit Intensity. For a real $s>0$ and a continuous function $f:[0,s]\to\mathbb{R}$, the Wiener integral $\int_0^{s}f(u)\,dB_u$ is the It^{o} integral of the family of constant random variables $(f(u))_{u\in(0,s]}$, which is It^{o} integrable by claim 3 of Adapted Mean-Square Continuous Processes are Ito Integrable; each Wiener integral is determined up to almost sure equality, and a fixed version is understood wherever one appears below.

1. For every real $s>0$, taking $f\equiv1$ gives $\int_0^{s}1\,dB_u=B_s$ almost surely.

2. Let $p\ge1$ be a natural number, let $s_1,\dots,s_p>0$ be real, and for each $i$ let $f_i:[0,s_i]\to\mathbb{R}$ be continuous. Then

is a Gaussian random vector whose expectations vanish and whose covariances are

the Riemann integral of the continuous product, which exists by claim 3 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval. (Since the covariances are unchanged when a random variable is replaced by an almost surely equal one, this holds for every choice of versions.)

3. Consequently, for any choice of versions, the family consisting of all Wiener integrals $\int_0^{s}f(u)\,dB_u$ (over all real $s>0$ and all continuous $f:[0,s]\to\mathbb{R}$) together with all the random variables $B_t$ ($t\ge0$) is jointly Gaussian.