Brownian Increments After a Time are Independent of the Model Past

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. Consider a linear-Gaussian state-observation model on $[0,T]$, with notation and fixed versions as there. Fix $s\in[0,T]$ and let $\mathcal{H}_s$ denote the $\sigma$-algebra generated by the combined family of the random variables $\xi^{i}$ ($1\le i\le l$), $W^{j}_r$ ($1\le j\le m$, $0\le r\le s$), and $u^{j}_r$ ($1\le j\le\tilde l$, $0\le r\le s$). Since the values of $u$ up to time $s$ are $\mathcal{H}_s$-measurable, $\mathcal{G}_s\subseteq\mathcal{H}_s$.

1. (Independence of later increments) For every $n\ge1$ and all choices of component indices $j_1,\dots,j_n\in\{1,\dots,m\}$ and times $s\le u_p<v_p\le T$ ($1\le p\le n$), the $\sigma$-algebras

$$\sigma\bigl(W^{j_1}_{v_1}-W^{j_1}_{u_1},\ \dots,\ W^{j_n}_{v_n}-W^{j_n}_{u_n}\bigr)\quad\text{and}\quad\mathcal{H}_s$$

are independent.

2. (Span structure of Wiener-integral increments) Let $j\in\{1,\dots,m\}$, let $0\le s<t\le T$, let $f:[0,t]\to\mathbb{R}$ be continuous, and set

$$I:=\int_0^t f(r)\,dW^{j}_r-\int_0^s f(r)\,dW^{j}_r ,$$

with Wiener integrals (any fixed versions; for $s=0$ the subtracted term is $0$ by the convention of Ito Integrable Process and the Ito Integral). Then $\mathbb{E}[I]=0$ with the expectation, and $I$ lies in the closed mean-square span of the family of increments $W^{j}_v-W^{j}_u$ with $s\le u<v\le t$.

3. (Orthogonality) With $I$ as in claim 2, for every square-integrable random variable $Q$ that is almost surely equal to an $\mathcal{H}_s$-measurable random variable: $IQ$ is integrable, $\mathbb{E}[IQ]=0$, and, with the covariance, $\operatorname{Cov}(I,Q)=0$.