Superposition Decomposition of the Controlled State and Observations

Consider a linear-Gaussian state-observation model on $[0,T]$, a control dimension $k\ge1$, a control matrix assignment $B$, an admissible control $\alpha$, and the controlled state and observations $X^{\alpha}$, $u^{\alpha}$, with all notation and fixed versions as in those items. Let $\Phi$ and $\Psi=\Phi^{-1}$ be the fundamental solution of $A$ on $[0,T]$ and its inverse from Variation of Constants for Linear Systems. Define, with fixed versions of the mean-square Riemann integrals, the correction processes

$$c_t:=\Phi(t)\,h_t,\qquad h^{j}_t:=\int_0^t\bigl(\Psi(r)B(r)\alpha_r\bigr)^{j}\,dr\quad(1\le j\le l),\qquad \gamma^{j}_t:=\int_0^t\bigl(\tilde E(r)c_r\bigr)^{j}\,dr\quad(1\le j\le\tilde l),$$

for $t\in[0,T]$, with the matrix-vector product formed componentwise; the integrals defining $h$ exist by Existence and Uniqueness of the Mean-Square Riemann Integral for Mean-Square Continuous Families and claims 1-2 of Basic Properties of the Mean-Square Riemann Integral, and those defining $\gamma$ exist by the same results together with the mean-square continuity of $c$ established in claim 1 below. Then:

1. (Regularity and dynamics of the correction) Each component family $(c^{i}_t)_{t\in[0,T]}$ is mean-square continuous, $c^{i}_0=0$ almost surely, and, componentwise and almost surely,

$$c_t=\int_0^t\bigl(A(r)c_r+B(r)\alpha_r\bigr)\,dr\qquad(0\le t\le T);$$

that is, $c$ is a mean-square solution of the linear stochastic differential equation with coefficient $A$, forcing $(B(r)\alpha_r)_r$, zero noise matrix, and zero initial value.

2. (Superposition) For every $t\in[0,T]$ and every $i\in\{1,\dots,l\}$: $X^{\alpha,i}_t=X^{i}_t+c^{i}_t$ almost surely.

3. (Adaptedness of the corrections) For every $t$, each $c^{i}_t$ and each $\gamma^{j}_t$ is almost surely equal to a $\mathcal{G}_t$-measurable square-integrable random variable.

4. (Observation shift) Each component family $(\gamma^{j}_t)_{t\in[0,T]}$ is mean-square continuous, and for every $t$ and $j$: $u^{\alpha,j}_t=u^{j}_t+\gamma^{j}_t$ almost surely.