The Kalman-Bucy Filter Equation and Its Solution

Consider a linear-Gaussian state-observation model on $[0,T]$, with notation and fixed versions as there.

1. (Covariance Riccati equation and gain) The matrix $P_0:=\bigl(\operatorname{Cov}(\xi^{i},\xi^{j})\bigr)_{1\le i,j\le l}$, with the covariance (defined and finite by Square-Integrability, Moments, and Covariance Matrix of a Gaussian Random Vector), is symmetric positive semidefinite. The assignment $D(t):=\tilde E(t)^{\top}\tilde\Theta(t)^{-1}\tilde E(t)$ has continuous entries (Continuity of the Inverse of a Continuous Matrix Function) and every $D(t)$ is symmetric positive semidefinite. Hence, by Global Existence and Uniqueness for the Kalman Covariance Riccati Equation, there is exactly one assignment $\Pi$ with continuous entries satisfying

and every $\Pi(t)$ is symmetric positive semidefinite. The gain $K(t):=\Pi(t)\tilde E(t)^{\top}\tilde\Theta(t)^{-1}$ has continuous entries.

2. (The filter process) Call a family $n=(n_t)_{t\in[0,T]}$ a solution of the Kalman-Bucy filter equation if it is a mean-square solution of the linear stochastic differential equation with coefficient $A-K\tilde E$, forcing $g_r=K(r)\tilde E(r)X_r$ (componentwise mean-square continuous by claims 1-2 of Basic Properties of the Mean-Square Riemann Integral), noise matrix $K\tilde\varepsilon$, and constant initial value $\mathbb{E}[\xi]:=(\mathbb{E}[\xi^{1}],\dots,\mathbb{E}[\xi^{l}])$; equivalently, by the definition of the observation integral in Integrals Against the Observation Process are Determined by the Observations and linearity, if $n$ is componentwise mean-square continuous, square-integrable, and satisfies, componentwise and almost surely,

with the degenerate-time convention $\int_0^0K\,du=0$ of Integrals Against the Observation Process are Determined by the Observations. By Existence, Uniqueness, and Variation of Constants for Linear Stochastic Differential Equations, a solution exists — the filter process $m^{\mathrm f}=(m^{\mathrm f}_t)_{t\in[0,T]}$, given by claim 1 there with fixed versions — and any two solutions agree almost surely at each time. The forcing involves the unobserved state $X$; claim 3 is what makes the filter process determined by the observations.

3. (Adaptedness) For every $t$ and $i$, the random variable $(m^{\mathrm f}_t)^{i}$ is almost surely equal to a $\mathcal{G}_t$-measurable square-integrable random variable, and it is a mean-square limit of finite linear combinations of the constant $1$ and of the values $u^{j}_r$ ($1\le j\le\tilde l$, $0\le r\le t$).