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

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 $(A,g,\varepsilon,\xi,W)$ and $(A,g',\varepsilon,\xi',W)$ be linear stochastic differential equations with additive Wiener noise on $[0,T]$ that share the coefficient family $A$, the noise coefficient family $\varepsilon$, and the Brownian motion $W$, and let $X$ and $Y$ be 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

$$c_A=\sup\big\{|A_{ij}(t)|:t\in[0,T],\ i,j\in\{1,\dots,l\}\big\},$$

which is finite because each entry of $A$ is continuous on $[0,T]$ and hence bounded. Write $\mathbb{E}$ for the 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 exponential function. Then:

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

$$\mathbb{E}\big[|X_t-Y_t|^2\big]\ \le\ \Big(\mathbb{E}\big[|\xi-\xi'|^2\big]+\int_0^t\mathbb{E}\big[|g_r-g'_r|^2\big]\,dr\Big)\,\exp\big((2\,l\,c_A+1)\,t\big),$$

where the integral is the 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 mean-square continuity of expected bilinear forms, each component family $(g^i_r-g'^i_r)_{r\in[0,T]}$ being mean-square continuous as a difference of mean-square continuous families by the basic properties of mean-square limits and integrals, and hence Riemann integrable by claim 3 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval.

(b) (Stability along sequences.) Suppose that, for each 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 limits of real sequences,

$$\lim_{n\to\infty}\mathbb{E}\big[|\xi^{(n)}-\xi|^2\big]=0\qquad\text{and}\qquad\lim_{n\to\infty}\int_0^T\mathbb{E}\big[|g^{(n)}_r-g_r|^2\big]\,dr=0,$$

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

$$\lim_{n\to\infty}\ \sup\big\{\mathbb{E}\big[|X^{(n)}_t-X_t|^2\big]:t\in[0,T]\big\}=0 .$$