Proof of The Kalman-Bucy Filter Computes the Conditional Expectation in the Linear-Gaussian Model

Write $\bar A=A-K\tilde E$, let $\bar\Phi$ be the fundamental solution of $\bar A$ on $[0,T]$ (Fundamental Solution and Variation of Constants for Linear Ordinary Differential Equations), and put $\xi_c=\xi-\mathbb{E}[\xi]$ (componentwise; $\xi_c^{i}=\xi^{i}-\mathbb{E}[\xi^{i}]$). All matrix-entry and transpose manipulations use claims 1-3 of Componentwise Estimates, Transpose Identities, and Indefinite Riemann Integrals and Associativity of the Matrix Product; covariances and their bilinearity are from Covariance of Square-Integrable Random Variables; all identities between random variables are almost sure. Write $\mathcal{S}_s$ for the closed mean-square span of the collection consisting of the constant $1$, the components $\xi^{i'}$, and the values $W^{j'}_r$ with $0\le r\le s$.

Step 1 (error equation). Subtracting the filter equation (claim 2 of The Kalman-Bucy Filter Equation and Its Solution) from the state equation (Linear-Gaussian State-Observation Model with Mean-Square Solution of a Linear Stochastic Differential Equation with Additive Wiener Noise), and using linearity of the mean-square Riemann integral (claim 1 of Basic Properties of the Mean-Square Riemann Integral) and of the Wiener integral (claim 1 of Properties of the Ito Integral: Linearity, Isometry, Martingale Property, and Mean-Square Continuity), componentwise:

since $A(r)X_r-\bigl(\bar A(r)m^{\mathrm f}_r+K(r)\tilde E(r)X_r\bigr)=\bar A(r)(X_r-m^{\mathrm f}_r)$ (expand both sides entrywise). Thus $e$ is a mean-square solution of the linear stochastic differential equation with data $(\bar A,0,\varepsilon-K\tilde\varepsilon,\xi_c,W)$, and any mean-square solution of that equation agrees with $e$ almost surely at each time (claim 2 of Existence, Uniqueness, and Variation of Constants for Linear Stochastic Differential Equations). Note $\sigma(\xi_c^{1},\dots,\xi_c^{l})\subseteq\sigma(\xi^{1},\dots,\xi^{l})$ (constant shifts preserve generated $\sigma$-algebras, Sigma-Algebra Generated by Random Variables and Independence of Sigma-Algebras), so $\sigma(\xi_c\text{-components})$ is independent of the $W$-$\sigma$-algebra, and the family of the $\xi_c^{i}$ and the $W^{j}_t$ is jointly Gaussian (finite tuples are affine images of tuples from the model's base family, Affine Transformations of Gaussian Random Vectors are Gaussian).

Step 2 (Gaussian and span structure). By claim 3 of Existence, Uniqueness, and Variation of Constants for Linear Stochastic Differential Equations applied to $e$'s equation and claim 1 of The Closed Mean-Square Span of a Family of Random Variables (the $\xi_c^{i'}$ lie in $\mathcal{S}_s$), each $e^{i}_s$ lies in $\mathcal{S}_s$. The same holds for $X^{i}_s$ and $u^{j}_s$ (claim 1 of Gaussian and Span Structure of the Linear-Gaussian State-Observation Model) and for $(m^{\mathrm f}_s)^{i}$ (claim 3 of The Kalman-Bucy Filter Equation and Its Solution: its generating collection — the constant $1$ and $u$-values at times $\le s$ — lies in $\mathcal{S}_s$, so claim 1 of The Closed Mean-Square Span of a Family of Random Variables applies). Every member of the grand family of all $\xi^{i}$, $W^{j}_t$, $X^{i}_t$, $u^{j}_t$, $(m^{\mathrm f}_t)^{i}$, and $e^{i}_t$ is therefore a mean-square limit of finite affine combinations of members of the base family $\{\xi^{i}\}\cup\{W^{j}_t\}$ (the constant $1$ contributing the affine part), which is jointly Gaussian by the model hypothesis; by Mean-Square Limits of Affine Combinations Adjoin to a Jointly Gaussian Family, the grand family is jointly Gaussian.

Step 3 (error covariance; claim 2). By claims 2-3 of Gaussian Structure and Moment Equations for Linear Stochastic Differential Equations applied to $e$'s equation: $\mathbb{E}[e_t]=\bar\Phi(t)\,\mathbb{E}[\xi_c]=0$ componentwise, and $\Sigma(t):=\bigl(\operatorname{Cov}(e^{i}_t,e^{j}_t)\bigr)_{ij}$ is the unique solution of the linear matrix equation of Lyapunov Representation and Positive Semidefiniteness for Linear Matrix Equations with coefficient $\bar A$, inhomogeneity $(\varepsilon-K\tilde\varepsilon)(\varepsilon-K\tilde\varepsilon)^{\top}$, and initial value $\bigl(\operatorname{Cov}(\xi_c^{i},\xi_c^{j})\bigr)_{ij}$. Covariances are unchanged by constant shifts, so the initial value is $P_0$; and by model hypothesis (i) ($\varepsilon\tilde\varepsilon^{\top}=0$, hence also $\tilde\varepsilon\varepsilon^{\top}=(\varepsilon\tilde\varepsilon^{\top})^{\top}=0$),

On the other hand, $\Pi$ satisfies the same equation: using $K=\Pi\tilde E^{\top}\tilde\Theta^{-1}$, symmetry of $\Pi$ and of $\tilde\Theta^{-1}$ (claim 1 of The Kalman-Bucy Filter Equation and Its Solution, claim 2 of Continuity of the Inverse of a Continuous Matrix Function), and $D=\tilde E^{\top}\tilde\Theta^{-1}\tilde E$,

so that $\bar A\Pi+\Pi\bar A^{\top}+\Theta+K\tilde\Theta K^{\top}=A\Pi+\Pi A^{\top}-2\Pi D\Pi+\Theta+\Pi D\Pi=A\Pi+\Pi A^{\top}-\Pi D\Pi+\Theta$, which is the Riccati integrand; hence, by the Riccati equation of The Kalman-Bucy Filter Equation and Its Solution and linearity of the entrywise Riemann integral (Agreement of the Riemann and Lebesgue Integrals for Continuous Functions on a Closed Interval, Linearity and Monotonicity of the Lebesgue Integral), $\Pi$ satisfies the same linear matrix integral equation with the same initial value $P_0$. By the uniqueness in claim 1 of Lyapunov Representation and Positive Semidefiniteness for Linear Matrix Equations, $\Sigma\equiv\Pi$; this is claim 2.

Step 4 (orthogonality). Define, for $r\in[0,T]$, the matrices $M(r)=\bigl(\mathbb{E}[e^{i}_ru^{j}_r]\bigr)_{ij}$ ($l\times\tilde l$) and $N(r)=\bigl(\mathbb{E}[e^{i}_r(m^{\mathrm f}_r)^{j}]\bigr)_{ij}$ ($l\times l$). Since $e$ is centered (Step 3), covariances against $e^{i}$ coincide with these expectations.

Each pair $(e^{i},u^{j})$, $(e^{i},(m^{\mathrm f})^{j})$ is of the integral form of Second-Moment Evolution for Processes of Integral Form on $[0,T]$, with the respective data read off from Step 1, the observation equation, and the filter equation. The orthogonality hypothesis there holds by its claim 2, because for every $s$ all of $e^{i}_s$, $u^{j}_s$, $(m^{\mathrm f}_s)^{j}$ lie in $\mathcal{S}_s$ (Step 2), and $\sigma(\xi\text{-components})$ is independent of the $W$-$\sigma$-algebra. Hence, by claim 1 of Second-Moment Evolution for Processes of Integral Form (entrywise) and the expansion $\mathbb{E}[e_rX_r^{\top}]=\Sigma(r)+N(r)=\Pi(r)+N(r)$ (from $X=e+m^{\mathrm f}$ and bilinearity):

using $M(0)=\bigl(\mathbb{E}[\xi_c^{i}u^{j}_0]\bigr)=0$ ($u_0=0$ almost surely, claim 1 of Gaussian and Span Structure of the Linear-Gaussian State-Observation Model) and $N(0)=\bigl(\mathbb{E}[\xi_c^{i}]\,\mathbb{E}[\xi^{j}]\bigr)=0$ (the initial value of $m^{\mathrm f}$ is the constant tuple $\mathbb{E}[\xi]$, and constants factor out of expectations). Simplify with $(\varepsilon-K\tilde\varepsilon)\tilde\varepsilon^{\top}=-K\tilde\Theta=-\Pi\tilde E^{\top}$ (hypothesis (i) and $K\tilde\Theta=\Pi\tilde E^{\top}$) and $(\varepsilon-K\tilde\varepsilon)(K\tilde\varepsilon)^{\top}=-K\tilde\Theta K^{\top}=-\Pi\tilde E^{\top}K^{\top}$:

since $\bar A^{\top}+\tilde E^{\top}K^{\top}=A^{\top}$. The entries of $N$ and $M$ are continuous (claim 1 of Second-Moment Evolution for Processes of Integral Form). The map $F(r,\Xi)=\bar A(r)\Xi+\Xi A(r)^{\top}$ on $\mathbb{R}^{l^{2}}$ is composition continuous (Sum and Product Rules for One-Dimensional Derivatives and Continuity) and Lipschitz: with $\alpha^{*}$ bounding all entries of $\bar A$ and $A$ (Extreme Value Theorem on a Compact Interval), each entry of $F(r,\Xi)-F(r,\Xi')$ is bounded by $2l\alpha^{*}|\Xi-\Xi'|_{e}$ by claim 2 of Componentwise Estimates, Transpose Identities, and Indefinite Riemann Integrals, and claim 1 there converts to the Euclidean bound with constant $2l^{3}\alpha^{*}$. The zero assignment solves the same integral equation with initial value $0$; by the uniqueness in Global Existence and Uniqueness for Lipschitz Ordinary Differential Equations in Integral Form, $N\equiv0$. Then $M(r)=\int_0^r\bar AM\,dr'$, and the same uniqueness argument (with $F(r,\Xi)=\bar A(r)\Xi$) gives $M\equiv0$.

Finally fix $0\le r\le t$ and $j$, and consider $H(\tau)=\bigl(\mathbb{E}[e^{i}_\tau u^{j}_r]\bigr)_{i}$ for $\tau\in[r,T]$; if $r=t$ we are done by $M\equiv0$, so let $r<t\le T$. Restarted representation: for $\tau\in[r,T]$, claim 5 of Basic Properties of the Mean-Square Riemann Integral and the interval-splitting of Wiener integrals (Ito Integrable Process and the Ito Integral, claim 1 of Properties of the Ito Integral: Linearity, Isometry, Martingale Property, and Mean-Square Continuity) turn Step 1's equation into, componentwise,

which is of the form of Second-Moment Evolution for Processes of Integral Form on the interval $[r,T]$ with $y=e^{i}_r$, $\alpha=(\bar Ae)^{i}$, and the Wiener increments in the sense of the convention there (the integrands being restrictions of continuous functions on $[0,T]$, the extended-integrand integrals of that convention agree with the displayed increments by claim 3 of Wiener Integrals Against a Vector Brownian Motion are Jointly Gaussian and uniqueness of mean-square limits). Take $Z$ the constant process $Z_\tau\equiv u^{j}_r$ on $[r,T]$ ($z=u^{j}_r$, $\beta\equiv0$, $h\equiv0$). The orthogonality hypothesis on $[r,T]$ holds by claim 2 of Second-Moment Evolution for Processes of Integral Form: for $r\le s<\tau\le T$, $Z_s=u^{j}_r$ lies in $\mathcal{S}_r\subseteq\mathcal{S}_s$, and the condition on $Z$'s (vanishing) Wiener integrands is trivial. Claim 1 of Second-Moment Evolution for Processes of Integral Form on $[r,T]$ then gives, entrywise,

with $H(r)$ the $j$-th column of $M(r)=0$. By the uniqueness in Global Existence and Uniqueness for Lipschitz Ordinary Differential Equations in Integral Form applied on $[r,T]$ (the zero function solves the same equation, and the coefficient map is Lipschitz as above), $H\equiv0$ on $[r,T]$; in particular $\operatorname{Cov}(e^{i}_t,u^{j}_r)=\mathbb{E}[e^{i}_tu^{j}_r]=0$.

Step 5 (independence; claim 3). Fix $t$ and finitely many pairs $(j_1,r_1),\dots,(j_p,r_p)$ with $r_q\le t$. The tuple $(e^{1}_t,\dots,e^{l}_t,u^{j_1}_{r_1},\dots,u^{j_p}_{r_p})$ is a Gaussian random vector (Step 2, Jointly Gaussian Families of Random Variables and Gaussian Processes) with $\operatorname{Cov}(e^{i}_t,u^{j_q}_{r_q})=0$ (Step 4), so by Uncorrelated Jointly Gaussian Blocks are Independent the $\sigma$-algebras $\sigma(e^{1}_t,\dots,e^{l}_t)$ and $\sigma(u^{j_1}_{r_1},\dots,u^{j_p}_{r_p})$ are independent. Now fix $B\in\sigma(e^{1}_t,\dots,e^{l}_t)$ and let $\mathcal{D}=\{A'\in\mathcal{F}:P(A'\cap B)=P(A')P(B)\}$. $\mathcal{D}$ is a $\lambda$-system in the sense of Dynkin's Pi-Lambda Theorem: $\Omega\in\mathcal{D}$; if $A'\subseteq A''$ both lie in $\mathcal{D}$ then $P((A''\setminus A')\cap B)=P(A''\cap B)-P(A'\cap B)=\bigl(P(A'')-P(A')\bigr)P(B)=P(A''\setminus A')P(B)$; and for a nondecreasing sequence $A'_n\in\mathcal{D}$ with union $A'$, continuity from below of $P$ (countable additivity applied to the disjoint differences) gives $P(A'\cap B)=\lim P(A'_n\cap B)=\lim P(A'_n)P(B)=P(A')P(B)$. The collection $\mathcal{P}$ of finite intersections of sets of the form $(u^{j}_r)^{-1}(B')$ with $r\le t$ and $B'$ a Borel set is a $\pi$-system contained in $\mathcal{D}$ (each such intersection lies in a $\sigma$-algebra $\sigma(u^{j_1}_{r_1},\dots,u^{j_p}_{r_p})$ independent of $\sigma(e\text{-components})\ni B$), and $\sigma(\mathcal{P})=\mathcal{G}_t$ (it contains the generators of $\mathcal{G}_t$ and is contained in it, Sigma-Algebra Generated by Random Variables and Independence of Sigma-Algebras). By Dynkin's Pi-Lambda Theorem, $\mathcal{G}_t\subseteq\mathcal{D}$. As $B$ was arbitrary, the two $\sigma$-algebras are independent (Sigma-Algebra Generated by Random Variables and Independence of Sigma-Algebras). The covariance statement is Step 4.

Step 6 (conditional expectation; claim 1). Fix $i$ and $t$, and let $Y$ be a $\mathcal{G}_t$-measurable square-integrable random variable with $(m^{\mathrm f}_t)^{i}=Y$ almost surely (claim 3 of The Kalman-Bucy Filter Equation and Its Solution). For every $A'\in\mathcal{G}_t$, the indicator $\mathbf{1}_{A'}$ is $\mathcal{G}_t$-measurable, and $e^{i}_t$ is $\sigma(e^{1}_t,\dots,e^{l}_t)$-measurable, so $e^{i}_t$ and $\mathbf{1}_{A'}$ are independent (Step 5 and the closing remark of Sigma-Algebra Generated by Random Variables and Independence of Sigma-Algebras); both are integrable with integrable product (Square-Integrable Random Variables and the Mean-Square Inner Product), so Expectation of a Product of Independent Random Variables and Step 3 give

Thus $Y$ satisfies properties (i)-(iii) of Conditional Expectation of a Square-Integrable Random Variable for $X^{i}_t$ and $\mathcal{G}_t$: it is a conditional expectation of $X^{i}_t$ given $\mathcal{G}_t$. By the uniqueness recorded there, every conditional expectation of $X^{i}_t$ given $\mathcal{G}_t$ equals $Y$ almost surely, and $Y=(m^{\mathrm f}_t)^{i}$ almost surely. $\blacksquare$