Proof of Integrals Against the Controlled Observations and the Controlled Filter Equation

Throughout, we use the rule (as in the proof of Superposition Decomposition of the Controlled State and Observations) that mean-square continuous integrand families that are almost surely equal at each time have almost surely equal mean-square Riemann integrals, and linearity and additivity of those integrals (claims 1 and 5 of Basic Properties of the Mean-Square Riemann Integral).

Claim 1. The Wiener-integral parts of $\int_0^t f\,du^{\alpha}_r$ and of the observation integral $\int_0^t f\,du_r$ of Integrals Against the Observation Process are Determined by the Observations are the same fixed versions. For the mean-square Riemann parts, claim 2 of Superposition Decomposition of the Controlled State and Observations gives $X^{\alpha}_r=X_r+c_r$ almost surely for each $r$, so the integrand families $\bigl((f(r)\tilde E(r)X^{\alpha}_r)^{i}\bigr)_r$ and $\bigl((f(r)\tilde E(r)X_r)^{i}+(f(r)\tilde E(r)c_r)^{i}\bigr)_r$ are almost surely equal at each time; by the rule above and linearity, componentwise and almost surely,

which is claim 1 (for $t=0$ all terms vanish by convention).

Claim 2. By claim 4 of Superposition Decomposition of the Controlled State and Observations, $u^{\alpha,j}_{x_p}-u^{\alpha,j}_{x_{p-1}}=\bigl(u^{j}_{x_p}-u^{j}_{x_{p-1}}\bigr)+\bigl(\gamma^{j}_{x_p}-\gamma^{j}_{x_{p-1}}\bigr)$ almost surely, so the Riemann-Stieltjes sums split accordingly. By claim 1 of Integrals Against the Observation Process are Determined by the Observations, the sums formed with $u$ converge componentwise in mean square to $\int_0^t f\,du_r$. By claim 1 above, it remains to show that, componentwise,

By claim 5 of Basic Properties of the Mean-Square Riemann Integral, $\gamma^{j}_{x_p}-\gamma^{j}_{x_{p-1}}=\int_{x_{p-1}}^{x_p}(\tilde E(r)c_r)^{j}\,dr$ almost surely, and the target integral is the sum over $p$ of $\int_{x_{p-1}}^{x_p}\bigl(f(r)\tilde E(r)c_r\bigr)^{i}\,dr$. Hence, componentwise and almost surely,

Let $\lVert\cdot\rVert_2$ be the mean-square norm. Each function $r\mapsto\lVert(\tilde E(r)c_r)^{j}\rVert_{2}$ is continuous (claim 4 of Basic Properties of the Mean-Square Riemann Integral, the integrand family being mean-square continuous by claims 1-2 there), hence bounded on $[0,t]$ by some real $C_j$ (Extreme Value Theorem on a Compact Interval); put $C:=\max_{1\le j\le\tilde l}C_j$. The entries of $f$ are uniformly continuous on $[0,t]$ by Continuity on a Closed Interval Implies Uniform Continuity; write $\omega_n$ for the maximum over entries $(i,j)$ of the oscillation of $f_{ij}$ over subintervals of length $t/n$, so $\omega_n\to0$. By the triangle inequality and claims 4 and 6 of Basic Properties of the Mean-Square Riemann Integral,

which proves claim 2.

Claim 3. By claim 2 of The Kalman-Bucy Filter Equation and Its Solution, componentwise and almost surely,

and by claim 1 of Superposition Decomposition of the Controlled State and Observations, $c_t=\int_0^t\bigl(A(r)c_r+B(r)\alpha_r\bigr)\,dr$ almost surely. On the other hand, by claim 1 above with $f=K$,

Therefore, using linearity and the rule for almost surely equal integrands (all integrand families being mean-square continuous), componentwise and almost surely,

since $(A-K\tilde E)(m^{\mathrm f}+c)+B\alpha+K\tilde Ec=(A-K\tilde E)m^{\mathrm f}+Ac+B\alpha$ pointwise (componentwise algebra of the matrix-vector product). By the two displayed equations for $m^{\mathrm f}$ and $c$, the right-hand side equals $m^{\mathrm f}_t+c_t=\widehat X_t$, which is the controlled filter equation.

For the equivalence: unpacking the definition of $\int_0^t K\,du^{\alpha}_r$ and combining the two mean-square Riemann integrals by linearity, the controlled filter equation states exactly that, componentwise and almost surely,

which, together with the mean-square continuity of the components of $\widehat X$ (claim 1 of Conditional Expectation and Estimation Error of the Controlled State) and of the forcing family (claims 1-2 of Basic Properties of the Mean-Square Riemann Integral), says precisely that $\widehat X$ is a mean-square solution of the linear stochastic differential equation with coefficient $A-K\tilde E$, forcing $\bigl(K(r)\tilde E(r)X^{\alpha}_r+B(r)\alpha_r\bigr)_r$, noise matrix $K\tilde\varepsilon$, and constant initial value $\mathbb{E}[\xi]$ (the initial tuple is a constant, hence square-integrable, and the entries of $K\tilde\varepsilon$ are continuous). Uniqueness among componentwise mean-square continuous square-integrable families satisfying the displayed equation is claim 2 of Existence, Uniqueness, and Variation of Constants for Linear Stochastic Differential Equations applied to this equation. $\square$