Mean-Square Solution of a Linear Stochastic Differential Equation with Additive Wiener Noise

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. Let $(\Omega,\mathcal{F},P)$ be a probability space, let $T>0$ be real, let $l,m\ge1$ be natural numbers, and let $W=(W^{1},\dots,W^{m})$ be an $m$-dimensional Brownian motion on $(\Omega,\mathcal{F},P)$. Let $A$ assign to each $t\in[0,T]$ a real $l\times l$ matrix $A(t)$ and $\varepsilon$ a real $l\times m$ matrix $\varepsilon(t)$, all entries being continuous functions of $t$; let $g=(g_t)_{t\in[0,T]}$ be a family of tuples $g_t=(g^{1}_t,\dots,g^{l}_t)$ of square-integrable random variables such that each component family $(g^{i}_t)_{t\in[0,T]}$ is mean-square continuous; and let $\xi=(\xi^{1},\dots,\xi^{l})$ be a tuple of square-integrable random variables. The data $(A,g,\varepsilon,\xi,W)$ is called a linear stochastic differential equation with additive Wiener noise on $[0,T]$.

A family $X=(X_t)_{t\in[0,T]}$ of tuples $X_t=(X^{1}_t,\dots,X^{l}_t)$ of square-integrable random variables, each component family $(X^{i}_t)_{t\in[0,T]}$ being mean-square continuous on $[0,T]$, is a mean-square solution of this equation if for every $t\in[0,T]$ and every $i\in\{1,\dots,l\}$, almost surely,

where the first integral is the mean-square Riemann integral — the integrand family is mean-square continuous by claims 1 and 2 of Basic Properties of the Mean-Square Riemann Integral, so the integral exists by Existence and Uniqueness of the Mean-Square Riemann Integral for Mean-Square Continuous Families — and the last integrals are Wiener integrals against the components of $W$, with $\int_0^0:=0$.

Notation. The displayed system is abbreviated, componentwise, as

with the matrix-vector product read entrywise in the random components.