Existence, Uniqueness, and Variation of Constants for Linear Stochastic Differential Equations

Let $(A,g,\varepsilon,\xi,W)$ be a linear stochastic differential equation with additive Wiener noise on $[0,T]$, with dimensions $l,m$ as there, and let $\Phi$ and $\Psi=\Phi^{-1}$ be the fundamental solution of $A$ on $[0,T]$ and its inverse from Fundamental Solution and Variation of Constants for Linear Ordinary Differential Equations.

1. (Existence and variation of constants) Fix versions of the integrals below and define, for $t\in[0,T]$ and $i\in\{1,\dots,l\}$,

with mean-square Riemann integrals (existing 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 Wiener integrals. Then $X=(X_t)_{t\in[0,T]}$ is a mean-square solution of the equation.

2. (Uniqueness) Any two mean-square solutions $X$ and $\widetilde X$ satisfy $X^{i}_t=\widetilde X^{i}_t$ almost surely, for every $i$ and $t$.

3. (Span structure) For every $t\in[0,T]$ and every $i$, the random variable $X^{i}_t$ of claim 1 is a mean-square limit of finite linear combinations of the following random variables: the components $\xi^{j}$ ($1\le j\le l$); the values $g^{j}_r$ ($1\le j\le l$, $0\le r\le t$); and the values $W^{j'}_r$ ($1\le j'\le m$, $0\le r\le t$).