Linear-Gaussian State-Observation Model

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. Let $(\Omega,\mathcal{F},P)$ be a probability space, let $T>0$ be real, let $l,\tilde l,m\ge1$ be natural numbers, and let $W=(W^{1},\dots,W^{m})$ be an $m$-dimensional Brownian motion on $(\Omega,\mathcal{F},P)$. Let $A$ ($l\times l$), $\varepsilon$ ($l\times m$), $\tilde E$ ($\tilde l\times l$), and $\tilde\varepsilon$ ($\tilde l\times m$) assign real matrices to each $t\in[0,T]$, all entries being continuous functions of $t$, and let $\xi=(\xi^{1},\dots,\xi^{l})$ be a tuple of random variables such that the combined family of the $\xi^{i}$ and the $W^{j}_t$ ($1\le j\le m$, $t\ge0$), indexed by the disjoint union, is jointly Gaussian (each $\xi^{i}$ is then square-integrable by Square-Integrability, Moments, and Covariance Matrix of a Gaussian Random Vector), and $\sigma(\xi^{1},\dots,\xi^{l})$ is independent of $\sigma(W^{j}_t:1\le j\le m,\ t\ge0)$, with the generated $\sigma$-algebras. Suppose, with the matrix product and transpose:

(i) (uncorrelated noises) $\varepsilon(t)\,\tilde\varepsilon(t)^{\top}=0$ for every $t\in[0,T]$;

(ii) (nondegenerate observation noise) $\tilde\Theta(t):=\tilde\varepsilon(t)\,\tilde\varepsilon(t)^{\top}$ is positive definite for every $t\in[0,T]$.

The linear-Gaussian state-observation model with these data consists of:

The state $X=(X_t)_{t\in[0,T]}$: a fixed choice of versions of a mean-square solution of the linear stochastic differential equation with data $(A,0,\varepsilon,\xi,W)$, where $0$ denotes the forcing family all of whose members are the zero tuple.

The observation process $u=(u_t)_{t\in[0,T]}$, $u_t=(u^{1}_t,\dots,u^{\tilde l}_t)$: a fixed choice of versions of

where the first integral is the mean-square Riemann integral, the last are Wiener integrals, and integrals over the degenerate interval at $t=0$ are $0$ by the conventions of Mean-Square Riemann Integral of a Family of Random Variables and Ito Integrable Process and the Ito Integral.

The observation $\sigma$-algebras $\mathcal{G}_t:=\sigma\bigl(u^{j}_r:1\le j\le\tilde l,\ 0\le r\le t\bigr)$ ($0\le t\le T$), with the $\sigma$-algebra generated by a family of random variables. The $\sigma$-algebras $\mathcal{G}_t$ are determined by the fixed choice of versions of $u$; every statement about the model refers to this fixed choice.

We further write $\Theta(t):=\varepsilon(t)\,\varepsilon(t)^{\top}$.

Well-definedness. A mean-square solution $X$ exists and any two agree almost surely at each time, by Existence, Uniqueness, and Variation of Constants for Linear Stochastic Differential Equations; the integrand family of the first integral, $\bigl(\sum_{i}\tilde E_{ji}(r)X^{i}_r\bigr)_{r}$, is mean-square continuous by claims 1-2 of Basic Properties of the Mean-Square Riemann Integral, so that integral exists by Existence and Uniqueness of the Mean-Square Riemann Integral for Mean-Square Continuous Families.