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),

$$\Bigl(\int_0^t f(r)\,du^{\alpha}_r\Bigr)^{i}:=\int_0^t\bigl(f(r)\tilde E(r)X^{\alpha}_r\bigr)^{i}\,dr+\sum_{j'=1}^{m}\int_0^t\bigl(f(r)\tilde\varepsilon(r)\bigr)_{ij'}\,dW^{j'}_r ,$$

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,

$$\int_0^t f(r)\,du^{\alpha}_r=\int_0^t f(r)\,du_r+\int_0^t f(r)\bigl(\tilde E(r)c_r\bigr)\,dr ,$$

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,

$$\Bigl\lVert\sum_{p=1}^{n}\Bigl(f(x_{p-1})\bigl(u^{\alpha}_{x_p}-u^{\alpha}_{x_{p-1}}\bigr)\Bigr)^{i}-\Bigl(\int_0^t f(r)\,du^{\alpha}_r\Bigr)^{i}\Bigr\rVert_{2}\longrightarrow0\qquad(n\to\infty).$$

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

$$\widehat X_t=\mathbb{E}[\xi]+\int_0^t\Bigl(\bigl(A(r)-K(r)\tilde E(r)\bigr)\widehat X_r+B(r)\alpha_r\Bigr)\,dr+\int_0^t K(r)\,du^{\alpha}_r ,$$

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.