Proof of Superposition Decomposition of the Controlled State and Observations

Throughout, mean-square Riemann integrals exist by Existence and Uniqueness of the Mean-Square Riemann Integral for Mean-Square Continuous Families, and we use repeatedly that two mean-square continuous integrand families that are almost surely equal at each time have almost surely equal integrals: their Riemann sums are almost surely equal for each partition, and mean-square limits are determined up to almost sure equality (claim 1 of Existence and Uniqueness of the Mean-Square Riemann Integral for Mean-Square Continuous Families). Consequently, a family whose components are mean-square continuous and which agrees almost surely at each time with a mean-square solution of a linear stochastic differential equation is itself a mean-square solution of that equation: every integral appearing in the defining identity is unchanged, up to almost sure equality, when the integrand family is so modified.

Claim 1. For each $j$, the integrand family $\bigl((\Psi(r)B(r)\alpha_r)^{j}\bigr)_{r}=\bigl(\sum_{\kappa}(\Psi(r)B(r))_{j\kappa}\alpha^{\kappa}_r\bigr)_{r}$ is mean-square continuous by claims 1-2 of Basic Properties of the Mean-Square Riemann Integral, the entries of $\Psi B$ being continuous (products and sums of continuous functions, Sums and Products of Continuous Real-Valued Functions). By claim 6 of Basic Properties of the Mean-Square Riemann Integral, each $(h^{j}_t)_t$ is mean-square continuous, and then so is each $(c^{i}_t)_t=\bigl(\sum_j\Phi_{ij}(t)h^{j}_t\bigr)_t$ by claims 1-2 of the same lemma. At $t=0$, each $h^{j}_0=0$ by the degenerate-interval convention of Mean-Square Riemann Integral of a Family of Random Variables, so $c^{i}_0=0$ almost surely.

Consider now the linear stochastic differential equation with data $(A,g,0,0,W)$, where $g_r=B(r)\alpha_r$, the noise matrix is the zero $l\times m$ matrix, and the initial value is the zero tuple; its forcing is mean-square continuous as in the well-definedness paragraph of Controlled State and Controlled Observations in the Linear-Gaussian Model. In claim 1 of Existence, Uniqueness, and Variation of Constants for Linear Stochastic Differential Equations for these data, the Wiener-integral terms $Z^{j}_t$ are integrals of the zero function, which vanish almost surely: their second moment is zero, by claim 2 of Properties of the Ito Integral: Linearity, Isometry, Martingale Property, and Mean-Square Continuity together with claim 3 of Existence and Uniqueness of the Mean-Square Extension of the Elementary Stochastic Integral applied to the constant approximating sequence $0$, whose norm is $0$. The initial tuple is zero, so the expression of that claim is, at each time, almost surely equal to $c_t$ as defined here (the mean-square Riemann integrals there and here are versions of integrals of the same families). Hence, by the rule stated at the beginning, $c$ is a mean-square solution of this equation, which by the definition of a mean-square solution means precisely that, componentwise and almost surely,

Claim 2. We show that the family $X+c$ (componentwise sums of the fixed versions) is a mean-square solution of the controlled equation with data $(A,g,\varepsilon,\xi,W)$. Its components are mean-square continuous as sums of mean-square continuous families. By the definition of the model, $X$ is a mean-square solution of the uncontrolled equation: componentwise and almost surely, $X_t=\xi+\int_0^t A(r)X_r\,dr+\sum_{j'}\int_0^t\varepsilon_{\cdot j'}(r)\,dW^{j'}_r$. Adding the identity of claim 1 and using linearity of the mean-square Riemann integral (claim 1 of Basic Properties of the Mean-Square Riemann Integral) together with $A(r)X_r+A(r)c_r=A(r)(X_r+c_r)$ (componentwise algebra of the matrix-vector product) and the rule for integrals of almost surely equal integrand families stated above, we obtain, componentwise and almost surely,

Thus $X+c$ is a mean-square solution of the controlled equation. Since $X^{\alpha}$ is also such a solution, claim 2 of Existence, Uniqueness, and Variation of Constants for Linear Stochastic Differential Equations gives $X^{\alpha,i}_t=X^{i}_t+c^{i}_t$ almost surely for every $i$ and $t$.

Claim 3. For $t=0$ we have $c^{i}_0=0=\gamma^{j}_0$ almost surely by the degenerate-interval convention, and the constant $0$ is $\mathcal{G}_0$-measurable; so assume $t>0$ and fix such a $t$. For $r\in[0,t]$ and each $j$, the value $(\Psi(r)B(r)\alpha_r)^{j}=\sum_{\kappa}(\Psi(r)B(r))_{j\kappa}\alpha^{\kappa}_r$ is almost surely equal to a $\mathcal{G}_r$-measurable square-integrable random variable by condition (ii) of Admissible Control for the Linear-Gaussian State-Observation Model and closure under finite linear combinations (claim 2 of The Closed Mean-Square Span of a Family of Random Variables); and $\mathcal{G}_r\subseteq\mathcal{G}_t$, the generating family of $\mathcal{G}_r$ being a subfamily of that of $\mathcal{G}_t$, so this version is also $\mathcal{G}_t$-measurable. Choose such a $\mathcal{G}_t$-measurable version for each $r\in[0,t]$; the resulting family on $[0,t]$ is mean-square continuous (mean-square distances are unchanged under almost sure modification). By claim 3 of Existence and Uniqueness of the Mean-Square Riemann Integral for Mean-Square Continuous Families, its integral over $[0,t]$ has a $\mathcal{G}_t$-measurable square-integrable version, and by the rule stated at the beginning, this integral is almost surely equal to $h^{j}_t$. Hence each $h^{j}_t$, and then each $c^{i}_t=\sum_j\Phi_{ij}(t)h^{j}_t$, is almost surely equal to a $\mathcal{G}_t$-measurable square-integrable random variable, again by claim 2 of The Closed Mean-Square Span of a Family of Random Variables. The same argument applies to $\gamma^{j}_t=\int_0^t(\tilde E(r)c_r)^{j}\,dr$: for $r\le t$ the integrand value $(\tilde E(r)c_r)^{j}$ is almost surely equal to a $\mathcal{G}_r$-measurable, hence $\mathcal{G}_t$-measurable, square-integrable random variable by what was just proved, and the previous reasoning applies verbatim.

Claim 4. Mean-square continuity of $(\gamma^{j}_t)_t$ follows from claims 1-2 and 6 of Basic Properties of the Mean-Square Riemann Integral as in claim 1. By the definitions of $u$ (in the model) and $u^{\alpha}$ (in Controlled State and Controlled Observations in the Linear-Gaussian Model), the Wiener-integral terms of $u^{\alpha,j}_t$ and $u^{j}_t$ are the same fixed versions, so

By linearity (claim 1 of Basic Properties of the Mean-Square Riemann Integral), the right-hand side is almost surely the integral of the family $\bigl((\tilde E(r)(X^{\alpha}_r-X_r))^{j}\bigr)_r$, which by claim 2 is almost surely equal at each time to $\bigl((\tilde E(r)c_r)^{j}\bigr)_r$; by the rule for almost surely equal integrands, the integral is almost surely $\gamma^{j}_t$. Hence $u^{\alpha,j}_t=u^{j}_t+\gamma^{j}_t$ almost surely. $\square$