Proof of Wiener Integrals of Continuous Functions are Jointly Gaussian

Throughout, $(B,1)$ is the It^{o} integrator of Brownian Motion is an Ito Integrator with Unit Intensity, $\lVert\cdot\rVert_2$ is the mean-square norm of Square-Integrable Random Variables and the Mean-Square Inner Product, and $\lambda$ is Lebesgue measure.

Step 1 (Claim 1). The constant family $H_u=1$, $u\in(0,s]$, is a simple adapted process with representation $\bigl((0,s),(1)\bigr)$, so by the consistency remark of Ito Integrable Process and the Ito Integral, its It^{o} integral equals its elementary integral almost surely, and the latter is $1\cdot(B_s-B_0)$. Since $B_0=0$ almost surely (clause (i) of Standard Brownian Motion), $\int_0^s1\,dB_u=B_s$ almost surely.

Step 2 (Approximants for claim 2). Fix $p$, $s_1,\dots,s_p$, $f_1,\dots,f_p$ as in claim 2, and set $T^{*}=\max(s_1,\dots,s_p)$. For $k\ge1$ let $0=u^k_0<u^k_1<\dots<u^k_{m_k}=T^{*}$ enumerate the finite set $\{iT^{*}/k:0\le i\le k\}\cup\{s_1,\dots,s_p\}$; its mesh is at most $T^{*}/k$. Define, for each $i$, the simple adapted process $H^{i,k}$ on $(0,T^{*}]$ whose value on the cell $(u^k_l,u^k_{l+1}]$ is the constant $f_i(u^k_l)$ if $u^k_{l+1}\le s_i$ and $0$ otherwise (constants are measurable for every $\sigma$-algebra and square-integrable, so this is a representation in the sense of Simple Adapted Process).

We check that $(H^{i,k})_k$ is an approximating sequence, in the sense of Existence and Uniqueness of the Mean-Square Extension of the Elementary Stochastic Integral, for the deterministic family $g_i$ on $(0,T^{*}]$ given by $g_i(u)=f_i(u)$ for $u\le s_i$ and $g_i(u)=0$ for $u>s_i$. Condition (b): fix $u\in(0,T^{*}]$ and let $(u^k_l,u^k_{l+1}]$ be the cell containing $u$. If $u\le s_i$, then $u^k_{l+1}\le s_i$ (as $s_i$ is a grid point), so $\lVert H^{i,k}_u-g_i(u)\rVert_2=|f_i(u^k_l)-f_i(u)|\to0$ by continuity of $f_i$ and $|u^k_l-u|\le T^{*}/k\to0$. If $u>s_i$, then $u^k_l\ge s_i$, so $H^{i,k}_u=0=g_i(u)$. Condition (a): $f_i$ is uniformly continuous on $[0,s_i]$ by Continuity on a Closed Interval Implies Uniform Continuity; given $\varepsilon>0$ pick $\delta>0$ with $|f_i(x)-f_i(y)|^{2}<\varepsilon/(2T^{*})$ for $|x-y|<\delta$, $x,y\in[0,s_i]$. For indices $k',k$ with $2T^{*}/\min(k',k)<\delta$ and any $u$: either both $H^{i,k'}_u$ and $H^{i,k}_u$ equal values of $f_i$ at points of $[0,s_i]$ within $\delta$ of each other (both left endpoints lie in $[0,s_i]$ and within the meshes of $u$, when $u\le s_i$), or both vanish (when $u>s_i$); in either case $\mathbb{E}[(H^{i,k'}_u-H^{i,k}_u)^{2}]\le\varepsilon/(2T^{*})$, so by monotonicity of the Lebesgue integral (Linearity and Monotonicity of the Lebesgue Integral) the integral in condition (a) is at most $\varepsilon/2<\varepsilon$.

Write $Y^k_i=\int_0^{T^{*}}H^{i,k}_u\,dB_u$ (elementary) and let $Y_i$ be the mean-square limit provided by Existence and Uniqueness of the Mean-Square Extension of the Elementary Stochastic Integral. We claim $Y_i=\int_0^{s_i}f_i(u)\,dB_u$ almost surely. Indeed, the restriction of $H^{i,k}$ to $(0,s_i]$ is an approximating sequence for $(f_i(u))_{u\in(0,s_i]}$ (condition (b) as above; condition (a) by monotonicity, as in Ito Integrable Process and the Ito Integral), and the elementary integrals coincide: cells beyond $s_i$ carry coefficient $0$, so the refined sums defining $Y^k_i$ and $\int_0^{s_i}H^{i,k}_u\,dB_u$ are equal term by term. One sequence of elementary integrals thus converges in mean square to both $Y_i$ and $\int_0^{s_i}f_i(u)\,dB_u$, so these agree almost surely by the null-equivalence clause of Square-Integrable Random Variables and the Mean-Square Inner Product.

Step 3 (Stage-$k$ vectors are Gaussian). Each $Y^k_i=\sum_l c^k_{il}\,(B_{u^k_{l+1}}-B_{u^k_l})$ with deterministic coefficients $c^k_{il}$. By condition (a) of Gaussian Process Characterization of Standard Brownian Motion, $B$ is a Gaussian process, so the tuple $(B_{u^k_0},\dots,B_{u^k_{m_k}})$ is a Gaussian random vector; the map taking it to $(Y^k_1,\dots,Y^k_p)$ is linear with real coefficients, so $(Y^k_1,\dots,Y^k_p)$ is a Gaussian random vector by Affine Transformations of Gaussian Random Vectors are Gaussian.

Step 4 (Passage to the limit). Since $\lVert Y^k_i-Y_i\rVert_2\to0$ for each $i$, Mean-Square Limits of Gaussian Random Vectors are Gaussian shows that $(Y_1,\dots,Y_p)$ is a Gaussian random vector with $\mathbb{E}[Y_i]=\lim_k\mathbb{E}[Y^k_i]=0$ (claim 2 of Linearity, Mean Zero, and Isometry of the Elementary Stochastic Integral) and $\operatorname{Cov}(Y_i,Y_j)=\lim_k\operatorname{Cov}(Y^k_i,Y^k_j)$. Since the means vanish, $\operatorname{Cov}(Y^k_i,Y^k_j)=\mathbb{E}[Y^k_iY^k_j]$ (Covariance of Square-Integrable Random Variables), and the polarization identity (claim 3 of Linearity, Mean Zero, and Isometry of the Elementary Stochastic Integral, with $\rho\equiv1$) gives, with $m=\min(s_i,s_j)$,

(the integrand is deterministic, and a product cell contributes exactly when it lies in both $(0,s_i]$ and $(0,s_j]$; the integral of the step function is the displayed sum, using that Lebesgue measure assigns to an interval its length). Since $m\in\{s_1,\dots,s_p\}$ is itself a grid point, the cells with $u^k_{l+1}\le m$ partition $(0,m]$, and the displayed sum is a left-endpoint sum for the continuous function $f_if_j$ on $[0,m]$ (continuity of products by Sum and Product Rules for One-Dimensional Derivatives and Continuity) over the induced partition of $[0,m]$. A left-endpoint sum lies between the lower and upper sums of $f_if_j$ for that partition (each sampled value lies between the infimum and supremum on its cell), as does the Riemann integral $\int_0^{m}f_if_j$, which exists by Continuous Functions on a Closed Interval are Riemann Integrable. By uniform continuity of $f_if_j$ on $[0,m]$ (Continuity on a Closed Interval Implies Uniform Continuity), the difference between the upper and lower sums is at most $\omega_k\cdot m$, where $\omega_k$ bounds the oscillation of $f_if_j$ over subintervals of length $T^{*}/k$ and $\omega_k\to0$; hence

With the identification $Y_i=\int_0^{s_i}f_i\,dB$ almost surely from Step 2 and Almost Sure Modifications of Gaussian Random Vectors are Gaussian (means and covariances are likewise unchanged under almost sure modification, the mean-square distance being $0$), this proves claim 2 for every choice of versions.

Step 5 (Claim 3). Consider any finite subfamily of the stated family: Wiener integrals $\int_0^{s_i}f_i\,dB$ ($1\le i\le p$) and values $B_{t_1},\dots,B_{t_q}$. If the subfamily contains no Wiener integral and only times $t_l=0$, it consists of random variables almost surely equal to the constant $0$, and is a Gaussian random vector by the $m=0$ clause of Gaussian Random Vectors and Jointly Gaussian Random Variables together with Almost Sure Modifications of Gaussian Random Vectors are Gaussian. Otherwise, for $t_l>0$, claim 1 gives $B_{t_l}=\int_0^{t_l}1\,dB$ almost surely, and for $t_l=0$, $B_0=0$ almost surely. By claim 2 applied to the combined list of integrands (the $f_i$ and the constant functions $1$ on $[0,t_l]$ for the indices with $t_l>0$), the corresponding vector of Wiener integrals, extended by the constant $0$ in the positions with $t_l=0$, is a Gaussian random vector (appending constant coordinates preserves Gaussianity by Affine Transformations of Gaussian Random Vectors are Gaussian). The original subfamily agrees with this vector componentwise almost surely, hence is a Gaussian random vector by Almost Sure Modifications of Gaussian Random Vectors are Gaussian. Since every finite subfamily is a Gaussian random vector, the family is jointly Gaussian. $\square$