Mean-Square Continuous Dependence for Linear Stochastic Differential Equations Driven by the Same Brownian Motion

Let $(\Omega,\mathcal{F},P)$ be a \reftext{def:probability-space-random-variable-2026a}{probability space}, let $T>0$ be real, let $l,m\ge1$ be \reftext{def:natural-numbers-2026a}{natural numbers}, and let $(A,g,\varepsilon,\xi,W)$ and $(A,g',\varepsilon,\xi',W)$ be \reftext{def:linear-sde-mean-square-solution-2026a}{linear stochastic differential equations with additive Wiener noise} on $[0,T]$ that share the coefficient family $A$, the noise coefficient family $\varepsilon$, and the \reftext{def:vector-brownian-motion-2026a}{Brownian motion} $W$, and let $X$ and $Y$ be \reftext{def:linear-sde-mean-square-solution-2026a}{mean-square solutions} of the first and of the second equation, respectively. No independence between the initial conditions and $W$, and no Gaussianity of the initial conditions, is assumed. Set

which is finite because each entry of $A$ is \reftext{def:continuity-closed-interval-c54-2026b}{continuous} on $[0,T]$ and hence \reftext{lem:continuous-compact-interval-bounded-2026a}{bounded}. Write $\mathbb{E}$ for the \reftext{def:expectation-variance-2026a}{expectation}, write $|x_t|^2=\sum_{i=1}^{l}(x^i_t)^2$ for a tuple $x_t=(x^1_t,\dots,x^l_t)$ of random variables, and write $\exp$ for the \reftext{def:exponential-function-real-2026a}{exponential function}. Then:

\textbf{(a) (Comparison bound.)} For every $t\in[0,T]$,

where the integral is the \reftext{def:riemann-integrable-closed-interval-c54-2026b}{Riemann integral} (with the convention $\int_0^0:=0$) of the function $r\mapsto\mathbb{E}[|g_r-g'_r|^2]$, which is continuous on $[0,T]$ by the \reftext{lem:expected-quadratic-form-2026a}{mean-square continuity of expected bilinear forms}, each component family $(g^i_r-g'^i_r)_{r\in[0,T]}$ being \reftext{def:mean-square-continuous-process-2026a}{mean-square continuous} as a difference of mean-square continuous families by \reftext{lem:mean-square-riemann-integral-properties-2026a}{the basic properties of mean-square limits and integrals}, and hence \reftext{lem:continuous-implies-riemann-integrable-c54-2026b}{Riemann integrable}.

\textbf{(b) (Stability along sequences.)} Suppose that, for each \reftext{def:natural-numbers-2026a}{natural number} $n$, $(A,g^{(n)},\varepsilon,\xi^{(n)},W)$ is a linear stochastic differential equation with additive Wiener noise on $[0,T]$ with the same $A$, $\varepsilon$, and $W$, and that $X^{(n)}$ is a mean-square solution of it. If, in the sense of \reftext{def:limit-sequence-real-c54-2026a}{limits of real sequences},

each integral existing as a Riemann integral of a continuous function for the same reason as in (a), then