Controlled State and Controlled Observations in the Linear-Gaussian 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.

Consider a linear-Gaussian state-observation model on $[0,T]$, with notation and fixed versions as there; let $k\ge1$ be a natural number; let $B$ assign to each $t\in[0,T]$ a real $l\times k$ matrix $B(t)$ whose entries are continuous functions of $t$; and let $\alpha=(\alpha_t)_{t\in[0,T]}$ be an admissible control with values in $\mathbb{R}^{k}$ for the model.

The controlled system with these data consists of:

The controlled state $X^{\alpha}=(X^{\alpha}_t)_{t\in[0,T]}$: a fixed choice of versions of a mean-square solution of the linear stochastic differential equation with data $(A,g,\varepsilon,\xi,W)$, where $g$ is the forcing family $g_r:=B(r)\,\alpha_r$ ($0\le r\le T$), formed componentwise with the matrix-vector product.

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

where the first integral is the mean-square Riemann integral, the last are Wiener integrals for which we fix the same versions as fixed for the observation process $u$ of the model, 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 controlled observation $\sigma$-algebras $\mathcal{G}^{\alpha}_t:=\sigma\bigl(u^{\alpha,j}_r:1\le j\le\tilde l,\ 0\le r\le t\bigr)$ ($0\le t\le T$), with the $\sigma$-algebra generated by a family of random variables, determined by the fixed choice of versions of $u^{\alpha}$.

Well-definedness. Each component family of the forcing $g$ is mean-square continuous by claims 1-2 of Basic Properties of the Mean-Square Riemann Integral, so a mean-square solution 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 $\bigl((\tilde E(r)X^{\alpha}_r)^{j}\bigr)_{r}$ is mean-square continuous by the same claims, so the first integral exists by Existence and Uniqueness of the Mean-Square Riemann Integral for Mean-Square Continuous Families.