Integrals Against the Observation Process are Determined by the Observations

Consider a linear-Gaussian state-observation model on $[0,T]$, with notation and fixed versions as there. 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 $t\in(0,T]$, let $k\ge1$ be a natural number, and let $f$ assign to each $r\in[0,t]$ a real $k\times\tilde l$ matrix $f(r)$ with continuous entries. Define, for $1\le i\le k$ (fixed versions),

with the mean-square Riemann integral (the integrand family is mean-square continuous by claims 1-2 of Basic Properties of the Mean-Square Riemann Integral, and the integral exists by Existence and Uniqueness of the Mean-Square Riemann Integral for Mean-Square Continuous Families) and Wiener integrals; for the degenerate time we set $\int_0^0f(r)\,du_r:=0$ (the zero tuple), consistently with the conventions of Mean-Square Riemann Integral of a Family of Random Variables.

1. (Riemann-Stieltjes approximation) For $n\ge1$ put $x_p=pt/n$ ($0\le p\le n$). Then, componentwise, with $\lVert\cdot\rVert_2$ the mean-square norm of Square-Integrable Random Variables and the Mean-Square Inner Product,

the sums being formed componentwise from the tuples $u_{x_p}$.

2. (Determination by the observations) Each component of $\int_0^tf(r)\,du_r$ is almost surely equal to a $\mathcal{G}_t$-measurable square-integrable random variable, and is a mean-square limit of finite linear combinations of the values $u^{j}_r$ ($1\le j\le\tilde l$, $0\le r\le t$).