Extended Admissible Control for the Linear-Gaussian State-Observation Model

Consider a linear-Gaussian state-observation model on $[0,T]$, with notation and fixed versions as there, and let $k\ge1$ be a natural number. Adopt the notation $\mathcal{B}_{[0,T]}$, $\lambda_{[0,T]}$ of the restricted Lebesgue measure on $[0,T]$, and the distance $d$ between admissible controls of Almost-Everywhere Mean-Square Limits of Cauchy Sequences of Admissible Controls.

An extended admissible control with values in $\mathbb{R}^{k}$ for the model is a family $\alpha=(\alpha_t)_{t\in[0,T]}$, where each $\alpha_t=(\alpha^{1}_t,\dots,\alpha^{k}_t)$ is a tuple of square-integrable random variables on $(\Omega,\mathcal{F},P)$, such that:

(i) for every $t\in[0,T]$ and every $\kappa\in\{1,\dots,k\}$, the random variable $\alpha^{\kappa}_t$ is almost surely equal to a $\mathcal{G}_t$-measurable square-integrable random variable, where $\mathcal{G}_t$ is the observation $\sigma$-algebra of the model itself, without control, and measurability means that preimages of Borel sets belong to $\mathcal{G}_t$;

(ii) there exist a sequence $(\alpha^{(n)})_{n\in\mathbb{N}}$ of admissible controls with values in $\mathbb{R}^{k}$ and a set $D\in\mathcal{B}_{[0,T]}$ with $\lambda_{[0,T]}([0,T]\setminus D)=0$ such that the sequence is Cauchy for $d$ in the sense of Almost-Everywhere Mean-Square Limits of Cauchy Sequences of Admissible Controls, and for every $t\in D$ and every $\kappa$ the real sequence $\bigl(\lVert\alpha^{(n),\kappa}_t-\alpha^{\kappa}_t\rVert_{2}\bigr)_{n}$ has limit $0$, with the mean-square norm $\lVert\cdot\rVert_{2}$ of Square-Integrable Random Variables and the Mean-Square Inner Product.

A pair $\bigl((\alpha^{(n)})_{n\in\mathbb{N}},D\bigr)$ as in (ii) is called an approximating sequence for $\alpha$.