Integrals Against the Controlled Observations and the Controlled Filter Equation

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 $[0,T]$, a control dimension $k\ge1$, a control matrix assignment $B$, an admissible control $\alpha$, and the controlled state and observations $X^{\alpha}$, $u^{\alpha}$, with notation and fixed versions as in those items. Let $c$, $\gamma$ be the correction processes and $\widehat X$ the controlled estimator, with the fixed versions of Superposition Decomposition of the Controlled State and Observations and Conditional Expectation and Estimation Error of the Controlled State, and let $K$ and $m^{\mathrm f}$ be the gain and filter process of The Kalman-Bucy Filter Equation and Its Solution.

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 (existing by Existence and Uniqueness of the Mean-Square Riemann Integral for Mean-Square Continuous Families and claims 1-2 of Basic Properties of the Mean-Square Riemann Integral) and Wiener integrals with the versions fixed as in the model; set $\int_0^0 f(r)\,du^{\alpha}_r:=0$, consistently with Integrals Against the Observation Process are Determined by the Observations. Then:

1. (Decomposition) Componentwise and almost surely,

where the first integral on the right is the observation integral of Integrals Against the Observation Process are Determined by the Observations and the last is the mean-square Riemann integral (componentwise, with the matrix-vector product).

2. (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,

3. (Controlled filter equation) Componentwise and almost surely, for every $t\in[0,T]$,

with $\mathbb{E}[\xi]:=(\mathbb{E}[\xi^{1}],\dots,\mathbb{E}[\xi^{l}])$; equivalently, $\widehat X$ is a mean-square solution of the linear stochastic differential equation with coefficient $A-K\tilde E$, forcing family $\bigl(K(r)\tilde E(r)X^{\alpha}_r+B(r)\alpha_r\bigr)_r$, noise matrix $K\tilde\varepsilon$, and constant initial value $\mathbb{E}[\xi]$. Moreover, any family of $l$-tuples of square-integrable random variables whose components are mean-square continuous and which satisfies the displayed equation agrees with $\widehat X$ almost surely at each time.