A deterministic Itô integral has independent Gaussian increments

Fix $0\le s<t\le T$ and put $H(r)=g(r)1_{(s,t]}(r)$. By [F4], choose elementary predictable approximants $H_n(r,\omega )={\sum }_k{\xi }_{n,k}(\omega )1_{(r_{n,k},r_{n,k+1}]}(r)$ converging to $H$ in $L^2(dr\otimes P)$. On each interval define the deterministic coefficient ${\bar{\xi }}_{n,k}=E[{\xi }_{n,k}]$ and let ${\bar{H}}_n={\sum }_k{\bar{\xi }}_{n,k}{1}_{(r_{n,k},r_{n,k+1}]}$. Boundedness of the elementary coefficients makes these expectations finite. For every $r$ in the kth interval, nonnegativity of $\mathrm{Var}({\xi }_{n,k}-H(r))$ from [F6] gives $\mathrm{\mid }{\bar{\xi }}_{n,k}-H(r){\mathrm{\mid }}^2\le E\mathrm{\mid }{\xi }_{n,k}-H(r){\mathrm{\mid }}^2$. Integrating and applying [F7], with Tonelli [F8] to exchange the time integral and expectation of this nonnegative integrand, yields $\mathrm{\parallel }{\bar{H}}_n-H{\mathrm{\parallel }}_{L^2([0,T])}^2\le E{\int }_0^T\mathrm{\mid }{H}_n-H{\mathrm{\mid }}^{2}dr\to 0$. Set $H_n^0={\bar{H}}_n{1}_{(s,t]}$; it remains a deterministic step integrand and its error cannot increase, because $H=0$ outside $(s,t]$.

Write $H_n^0={\sum }_j{a}_{n,j}{1}_{(u_{n,j},u_{n,j+1}]}$ on a partition of $[s,t]$. By [F3], its vector integral is $Z_n={\sum }_j{a}_{n,j}(W_{u_{n,j+1}}-W_{u_{n,j}})$. For $\theta \in {\mathbb{R}}^d$, independence and the Gaussian increment laws in [F1] give $E{e}^{i\theta \cdot Z_n}={\prod }_j\exp (-a_{n,j}^2(u_{n,j+1}-u_{n,j})\mathrm{\mid }\theta {\mathrm{\mid }}^2\mathrm{/}2)=\exp (-\mathrm{\mid }\theta {\mathrm{\mid }}^2{\int }_s^t(H_n^0{)}^{2}dr\mathrm{/}2)$. The same factorization holds with any indicator $1_A$, $A\in {\mathcal{F}}_s$, inserted: each increment is independent of the sigma-algebra at its left endpoint, which contains $A$ and all preceding increments, so repeated factorization gives $E[1_A{e}^{i\theta \cdot Z_n}]=P(A)E{e}^{i\theta \cdot Z_n}$.

Let $Z=M_t-M_s$. By [F5] applied to the deterministic step integrands, $E\mathrm{\mid }{Z}_n-Z{\mathrm{\mid }}^2=d{\int }_s^t\mathrm{\mid }{H}_n^0(r)-g(r){\mathrm{\mid }}^{2}dr\to 0$. Since $\mathrm{\mid }{e}^{i\theta \cdot z}-e^{i\theta \cdot w}\mathrm{\mid }\le \mathrm{\mid }\theta \mathrm{\mid }\mathrm{\mid }z-w\mathrm{\mid }$, nonnegativity of the variance of $\mathrm{\mid }{Z}_n-Z\mathrm{\mid }$ in [F6] gives convergence of both the ordinary and $1_A$-weighted characteristic functions. Also ${\int }_s^t(H_n^0{)}^{2}dr\to {\int }_s^{t}g(r{)}^{2}dr=v(t)-v(s)$ by the square-integrable norm convergence in step 1.1 and the norm-continuity result [F7]. Passing to the limit in step 1.2 therefore gives $E{e}^{i\theta \cdot Z}=\exp (-\mathrm{\mid }\theta {\mathrm{\mid }}^2(v(t)-v(s))\mathrm{/}2)$ and $E[1_A{e}^{i\theta \cdot Z}]=P(A)\exp (-\mathrm{\mid }\theta {\mathrm{\mid }}^2(v(t)-v(s))\mathrm{/}2)$ for every $\theta$ and every $A\in {\mathcal{F}}_s$.

The multivariate-normal convention [F9] identifies the Gaussian law from its one-dimensional normal projections, and [F10] gives the characteristic function in step 2.1 for those projections. For any $A\in {\mathcal{F}}_s$ with $P(A)>0$, the probability measure $B\mapsto P(Z\in B,A)\mathrm{/}P(A)$ has the same characteristic function by the second identity, so uniqueness in [F10] identifies it with that same Gaussian law. If $P(A)=0$, the joint probability is zero. Thus $Z$ is independent of ${\mathcal{F}}_s$.

By [F11], $M$ has an adapted continuous version. For any finite increasing list $0=t_0<\cdots <t_n$, the increments through time $t_{j-1}$ are ${\mathcal{F}}_{t_{j-1}}$-measurable. Step 3.1 makes the next increment independent of that sigma-algebra; induction on $j$ therefore factors the joint probabilities of all increment Borel sets into the product of their marginal probabilities. This proves mutual independence over the list.