Almost-Everywhere Mean-Square Limits of Cauchy Sequences of Admissible Controls

Consider a linear-Gaussian state-observation model on $[0,T]$ with observation $\sigma$-algebras $\mathcal{G}_t$, let $k\ge1$ be a natural number, and let $(\alpha^{(n)})_{n\in\mathbb{N}}$ be a sequence of admissible controls with values in $\mathbb{R}^{k}$ for the 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. For admissible controls $\beta,\gamma$ define, with the mean-square norm $\lVert\cdot\rVert_{2}$ of Square-Integrable Random Variables and the Mean-Square Inner Product,

$$d(\beta,\gamma):=\Bigl(\int_0^T\sum_{\kappa=1}^{k}\lVert\beta^{\kappa}_t-\gamma^{\kappa}_t\rVert_{2}^{2}\,dt\Bigr)^{1/2},$$

a Riemann integral of a function of $t$ that is continuous, and hence integrable by claim 3 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval: the componentwise difference family is mean-square continuous by claim 1 of Basic Properties of the Mean-Square Riemann Integral, so continuity of the integrand follows from claim 2 of Expected Bilinear Forms: Trace Formula and Mean-Square Continuity applied to that family and the identity matrix. Suppose the sequence is Cauchy for $d$: for every real $\varepsilon>0$ there is $N\in\mathbb{N}$ with $d(\alpha^{(n)},\alpha^{(m)})<\varepsilon$ for all $n,m\ge N$. Adopt the notation $\mathcal{B}_{[0,T]}$, $\lambda_{[0,T]}$ of the restricted Lebesgue measure on $[0,T]$, and call $D\in\mathcal{B}_{[0,T]}$ co-null if $\lambda_{[0,T]}([0,T]\setminus D)=0$. Then:

1. (Existence of an almost-everywhere limit) There exist a co-null set $D\in\mathcal{B}_{[0,T]}$, natural numbers $n_1<n_2<n_3<\dots$, and a family $\beta=(\beta_t)_{t\in[0,T]}$ of tuples $\beta_t=(\beta^{1}_t,\dots,\beta^{k}_t)$ of square-integrable random variables such that: $\beta^{\kappa}_t=0$ for every $t\notin D$ and every $\kappa$; for every $t\in[0,T]$ and $\kappa$ the random variable $\beta^{\kappa}_t$ is $\mathcal{G}_t$-measurable (preimages of Borel sets belong to $\mathcal{G}_t$); and for every $t\in D$ and every $\kappa$ the real sequence $\bigl(\lVert\alpha^{(n_j),\kappa}_t-\beta^{\kappa}_t\rVert_{2}\bigr)_{j}$ has limit $0$.

2. (Integrated convergence of the full sequence) Let $\beta$ be any family of tuples of square-integrable random variables and $D\in\mathcal{B}_{[0,T]}$ a co-null set such that, for some natural numbers $n_1<n_2<\dots$, every $t\in D$, and every $\kappa$: $\lVert\alpha^{(n_j),\kappa}_t-\beta^{\kappa}_t\rVert_{2}\to0$ as $j\to\infty$. Then for every $n$ the function

$$t\mapsto\mathbf{1}_D(t)\sum_{\kappa=1}^{k}\lVert\alpha^{(n),\kappa}_t-\beta^{\kappa}_t\rVert_{2}^{2}$$

is $\mathcal{B}_{[0,T]}$-measurable with finite Lebesgue integral, and these integrals tend to $0$ as $n\to\infty$ along the full sequence.

3. (Uniqueness almost everywhere) If $(\beta,D)$ and $(\beta',D')$ both satisfy the hypotheses of claim 2 (with possibly different subsequences), then there is a co-null set $D''\in\mathcal{B}_{[0,T]}$ such that for every $t\in D''$ and every $\kappa$: $\beta^{\kappa}_t=\beta'^{\kappa}_t$ almost surely.