Proof of The Kalman-Bucy Filter Equation and Its Solution

Claim 1. The covariances $\operatorname{Cov}(\xi^{i},\xi^{j})$ are defined and finite by Square-Integrability, Moments, and Covariance Matrix of a Gaussian Random Vector (each finite subfamily of the $\xi^{i}$ is a Gaussian random vector by the model hypothesis and Jointly Gaussian Families of Random Variables and Gaussian Processes). $P_0$ is symmetric by symmetry of the covariance, and positive semidefinite: for $x\in\mathbb{R}^{l}$, bilinearity of the covariance (Covariance of Square-Integrable Random Variables) gives $x\cdot(P_0x)=\operatorname{Cov}\bigl(\sum_ix^{i}\xi^{i},\sum_jx^{j}\xi^{j}\bigr)=\operatorname{Var}\bigl(\sum_ix^{i}\xi^{i}\bigr)\ge0$, the variance being nonnegative.

$\tilde\Theta=\tilde\varepsilon\tilde\varepsilon^{\top}$ has continuous entries (Sum and Product Rules for One-Dimensional Derivatives and Continuity), is symmetric ($(UV)^{\top}=V^{\top}U^{\top}$, claim 3 of Componentwise Estimates, Transpose Identities, and Indefinite Riemann Integrals), and is positive definite by model hypothesis (ii); by Continuity of the Inverse of a Continuous Matrix Function, $t\mapsto\tilde\Theta(t)^{-1}$ has continuous entries and each $\tilde\Theta(t)^{-1}$ is symmetric positive definite. Then $D=\tilde E^{\top}\tilde\Theta^{-1}\tilde E$ has continuous entries (products of continuous), is symmetric ($D^{\top}=\tilde E^{\top}(\tilde\Theta^{-1})^{\top}\tilde E=D$), and is positive semidefinite: by the transpose-dot identity (claim 3 of Componentwise Estimates, Transpose Identities, and Indefinite Riemann Integrals), $x\cdot(Dx)=(\tilde Ex)\cdot\bigl(\tilde\Theta^{-1}(\tilde Ex)\bigr)\ge0$. Likewise $\Theta=\varepsilon\varepsilon^{\top}$ is symmetric with continuous entries and positive semidefinite ($x\cdot(\Theta x)=(\varepsilon^{\top}x)\cdot(\varepsilon^{\top}x)\ge0$). Hence Global Existence and Uniqueness for the Kalman Covariance Riccati Equation applies with data $(A,\Theta,D,P_0)$ on $[0,T]$ and yields the unique continuous, symmetric, positive semidefinite $\Pi$; and $K=\Pi\tilde E^{\top}\tilde\Theta^{-1}$ has continuous entries.

Claim 2. The data $(A-K\tilde E,\ g,\ K\tilde\varepsilon,\ \mathbb{E}[\xi],\ W)$ with $g_r=K(r)\tilde E(r)X_r$ satisfy the requirements of Mean-Square Solution of a Linear Stochastic Differential Equation with Additive Wiener Noise: the coefficient and noise matrices have continuous entries (products of continuous), the constant tuple $\mathbb{E}[\xi]$ is square-integrable, and each component family $(g^{i}_r)_r=\bigl(\sum_j(K\tilde E)_{ij}(r)X^{j}_r\bigr)_r$ is mean-square continuous by claims 1-2 of Basic Properties of the Mean-Square Riemann Integral. By Existence, Uniqueness, and Variation of Constants for Linear Stochastic Differential Equations, a mean-square solution of this equation exists — the filter process $m^{\mathrm f}$ of the statement — and any two agree almost surely at each time. The equivalence of the two characterizations in the statement is term-by-term: by the definition of $\int_0^tK\,du$ in Integrals Against the Observation Process are Determined by the Observations with $f=K$ restricted to $[0,t]$, its time part is $\int_0^t(K\tilde EX)^{i}\,dr$ and its Wiener part is $\sum_{j'}\int_0^t(K\tilde\varepsilon)_{ij'}\,dW^{j'}$, exactly the corresponding terms of the SDE form after splitting the drift integral by claim 1 of Basic Properties of the Mean-Square Riemann Integral; at $t=0$ both forms reduce to $\mathbb{E}[\xi]$ by the degenerate-time conventions. Hence a family solves the SDE if and only if it satisfies the displayed filter equation, and existence and almost-sure uniqueness transfer.

Claim 3. Fix $t$; for $t=0$, $m^{\mathrm f}_0=\mathbb{E}[\xi]$ is constant and the claim is trivial, so let $t>0$. Let $\bar\Phi$, $\bar\Psi=\bar\Phi^{-1}$ be the fundamental solution of $A-K\tilde E$ and its inverse (Fundamental Solution and Variation of Constants for Linear Ordinary Differential Equations). By claim 1 of Existence, Uniqueness, and Variation of Constants for Linear Stochastic Differential Equations, componentwise and almost surely,

The middle and last groups are, by claim 1 of Basic Properties of the Mean-Square Riemann Integral and linearity of the Wiener integral (claim 1 of Properties of the Ito Integral: Linearity, Isometry, Martingale Property, and Mean-Square Continuity), almost surely equal to the components of $\int_0^t\bigl(\bar\Psi K\bigr)(r)\,du_r$ in the sense of Integrals Against the Observation Process are Determined by the Observations (the matrix function $\bar\Psi K$ restricted to $[0,t]$ has continuous entries): the time parts match because $g_r=K\tilde EX_r$ gives $\bar\Psi(r)g_r=\bigl(\bar\Psi K\bigr)(r)\tilde E(r)X_r$, and the Wiener parts match by associativity of the matrix products (Associativity of the Matrix Product). By claim 2 of Integrals Against the Observation Process are Determined by the Observations, each component of $\int_0^t(\bar\Psi K)\,du$ is almost surely equal to a $\mathcal{G}_t$-measurable square-integrable random variable and lies in the closed mean-square span of the values $u^{j}_r$ ($r\le t$). Hence $(m^{\mathrm f}_t)^{i}$, an affine combination of such components with the constants $\bar\Phi_{ij}(t)\mathbb{E}[\xi^{j}]$, is almost surely equal to a $\mathcal{G}_t$-measurable square-integrable random variable (claim 2 of The Closed Mean-Square Span of a Family of Random Variables: linear combinations of $\mathcal{G}_t$-measurable random variables are $\mathcal{G}_t$-measurable, and constants are $\mathcal{G}_t$-measurable) and lies in the closed mean-square span of the constant $1$ and the values $u^{j}_r$ with $r\le t$ (claim 1 of The Closed Mean-Square Span of a Family of Random Variables), which is the final assertion. $\blacksquare$