Proof of 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. Throughout, write $D_t=X_t-Y_t$ componentwise ($D^i_t=X^i_t-Y^i_t$), $f_r=g_r-g'_r$, and $u(t)=\mathbb{E}\big[|D_t|^2\big]$, with the expectation; for real $l$-vectors $v,v'$ write $v\cdot v'=\sum_{i=1}^{l}v^iv'^i$, and set $\kappa=2\,l\,c_A+1$. Adopt the data and notation of the statement.

Step 1: the difference is a mean-square solution with zero noise coefficient. The families $(X^i_t)_{t\in[0,T]}$ and $(Y^i_t)_{t\in[0,T]}$ are square-integrable and mean-square continuous by the definition of a mean-square solution; hence each $D^i_t$ is square-integrable, being a difference of square-integrable random variables, and each family $(D^i_t)_{t\in[0,T]}$ is mean-square continuous by claim 1 of the basic properties of the mean-square Riemann integral. Fix $t\in[0,T]$ and $i\in\{1,\dots,l\}$. By the definition of a mean-square solution, almost surely

$$X^i_t=\xi^i+\int_0^t\Big(\sum_{j=1}^{l}A_{ij}(r)X^j_r+g^i_r\Big)dr+\sum_{j=1}^{m}\int_0^t\varepsilon_{ij}(r)\,dW^j_r,$$

and likewise for $Y^i_t$ with $\xi'$, $Y$, $g'$ in place of $\xi$, $X$, $g$. The Wiener-integral terms in the two identities may be taken to be the same random variables: any two versions of the same Ito integral agree almost surely, by claims 1 and 2 of the existence and uniqueness of the mean-square extension of the elementary stochastic integral (claim 2 identifying, up to almost-sure equality, the mean-square limits of any two approximating sequences). Subtracting, and combining the two mean-square Riemann integrals by linearity (claims 1 and 2 of the properties lemma),

$$D^i_t=(\xi^i-\xi'^i)+\int_0^t\Big(\sum_{j=1}^{l}A_{ij}(r)D^j_r+f^i_r\Big)dr\qquad\text{almost surely}.$$

The integrand family is mean-square continuous: each $(f^i_r)_{r\in[0,T]}$ is mean-square continuous by claim 1 of the properties lemma applied to the mean-square continuous families $(g^i_r)$ and $(g'^i_r)$ of the two equations, each $(A_{ij}(r)D^j_r)_{r\in[0,T]}$ is mean-square continuous by claim 2 (a continuous scalar function times a mean-square continuous family), and finite sums preserve mean-square continuity by claim 1. Moreover, for each $j$ and $t$, the Wiener integral $\int_0^t0\,dW^j$ has expectation $0$ and variance $\int_0^t0^2\,du=0$ — for $t>0$ by claims 2 and 3 of that theorem, and trivially for $t=0$, the integral being $0$ by the convention of the Ito integral — hence its second moment, the variance plus the squared mean, is $0$, and it vanishes almost surely by the null-equivalence clause of the mean-square framework. The zero $l\times m$ matrix family has continuous entries, so $(A,f,0,\xi-\xi',W)$ is a linear stochastic differential equation with additive Wiener noise, and the display exhibits $D$ as a mean-square solution of it.

Step 2: second-moment evolution. For $i\in\{1,\dots,l\}$ let $\hat{D}^i_t$ denote the right-hand side of the last display with fixed versions of the integrals, so that $D^i_t=\hat{D}^i_t$ almost surely for each $t$. Fix $i,i'\in\{1,\dots,l\}$ and apply claim 1 of the second-moment evolution lemma with $a=0$ and $b=T$ (so that its hypothesis $0\le a<b$ holds, as $T>0$), with starts $y=\xi^i-\xi'^i$ and $z=\xi^{i'}-\xi'^{i'}$, drift families $\alpha_r=\sum_{j}A_{ij}(r)D^j_r+f^i_r$ and $\beta_r=\sum_{j}A_{i'j}(r)D^j_r+f^{i'}_r$ (mean-square continuous by Step 1), and noise coefficients $f_j=h_j=0$; the symbols $a$, $b$, $y$, $z$, $Y$, $Z$, $\alpha$, $\beta$, $f_j$, $h_j$ of that lemma are placeholders, unrelated to the solution $Y$ and the drift difference $f$ of the present statement. With these data the lemma's processes are $Y_t=\hat{D}^i_t+\sum_{j=1}^{m}\int_0^t0\,dW^j_r$ and $Z_t=\hat{D}^{i'}_t+\sum_{j=1}^{m}\int_0^t0\,dW^j_r$, almost surely equal to $D^i_t$ and $D^{i'}_t$ by Step 1. The orthogonality hypothesis holds: each increment $\int_0^t0\,dW^j-\int_0^s0\,dW^j$ is $0$ almost surely (Step 1), and the covariance of an almost-surely-zero square-integrable random variable with any square-integrable random variable is $0$. Since the hypothesis and the conclusion of that lemma involve the processes only through covariances and expectations at fixed times, which are unchanged when a random variable is replaced by an almost-surely-equal square-integrable one, the resulting identity reads, for $t\in[0,T]$,

$$\mathbb{E}\big[D^i_tD^{i'}_t\big]=\mathbb{E}\big[(\xi^i-\xi'^i)(\xi^{i'}-\xi'^{i'})\big]+\int_0^t\Big(\mathbb{E}\big[\alpha_rD^{i'}_r\big]+\mathbb{E}\big[D^i_r\beta_r\big]\Big)dr,$$

with $t\mapsto\mathbb{E}[D^i_tD^{i'}_t]$ and the integrand continuous on $[0,T]$. Take $i'=i$ and sum over $i\in\{1,\dots,l\}$: the finitely many terms combine by linearity of the expectation (linearity of the Lebesgue integral) and by linearity of the Riemann integrals of continuous functions, each of which agrees with the corresponding Lebesgue integral by the Riemann--Lebesgue agreement of claim 3 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval. This gives

$$u(t)=u(0)+\int_0^t2\Big(\mathbb{E}\big[D_r\cdot A(r)D_r\big]+\mathbb{E}\big[D_r\cdot f_r\big]\Big)dr\qquad(t\in[0,T]),$$

with $u$ continuous on $[0,T]$; moreover $u(0)=\mathbb{E}\big[|\xi-\xi'|^2\big]$, by the solution identity of Step 1 at $t=0$ (both integrals vanish there, with the convention $\int_0^0:=0$) and equality of second moments of almost-surely-equal square-integrable random variables.

Step 3: estimate and Gronwall. Pointwise on $\Omega$,

$$\big|D_r\cdot A(r)D_r\big|\le c_A\Big(\sum_{i=1}^{l}|D^i_r|\Big)^{2}\le c_A\,l\,|D_r|^2,\qquad\qquad 2\,\big|D_r\cdot f_r\big|\le2\sum_{i=1}^{l}|D^i_r|\,|f^i_r|\le\sum_{i=1}^{l}\big((D^i_r)^2+(f^i_r)^2\big)=|D_r|^2+|f_r|^2,$$

by elementary inequalities, each an instance of $2\lambda\mu\le\lambda^2+\mu^2$ summed over indices. Hence, by linearity and monotonicity of the expectation, $2\big(\mathbb{E}[D_r\cdot A(r)D_r]+\mathbb{E}[D_r\cdot f_r]\big)\le\kappa\,u(r)+\mathbb{E}[|f_r|^2]$ for every $r$, and monotonicity of the integral — via the Riemann--Lebesgue agreement of claim 3 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval for the continuous integrands and monotonicity of the Lebesgue integral — gives

$$u(t)\le u(0)+\int_0^t\mathbb{E}\big[|f_r|^2\big]\,dr+\kappa\int_0^tu(r)\,dr\qquad(t\in[0,T]),$$

the continuity of $r\mapsto\mathbb{E}[|f_r|^2]$ being recorded in clause (a) of the statement. Fix $t_0\in[0,T]$. For $t_0=0$, clause (a) at $t=0$ asserts $u(0)\le\mathbb{E}[|\xi-\xi'|^2]$ (with $\int_0^0:=0$ and $\exp(0)=1$, claim 1 of the properties of the exponential function), which holds with equality by Step 2. For $t_0>0$ set $a_0=u(0)+\int_0^{t_0}\mathbb{E}[|f_r|^2]\,dr$. For $t\in[0,t_0]$ the middle term above is at most $\int_0^{t_0}\mathbb{E}[|f_r|^2]\,dr$ — by additivity of the Riemann integral over adjacent intervals together with monotonicity of the integral as above and nonnegativity of the continuous integrand, the cases $t=0$ and $t=t_0$ being covered by the convention $\int_0^0:=0$ — so $u(t)\le a_0+\kappa\int_0^tu(r)\,dr$ on $[0,t_0]$, and Gronwall's lemma on $[0,t_0]$, applied to the continuous function $u$ with the constants $a_0$ and $\kappa\ge0$, gives $u(t)\le a_0\exp(\kappa\,t)$ on $[0,t_0]$. Evaluating at $t=t_0$ and renaming $t_0$ as $t$ yields clause (a).

Step 4: stability (clause (b)). For each natural number $n$, Steps 1--3 applied to the mean-square solutions $X^{(n)}$ and $X$ of $(A,g^{(n)},\varepsilon,\xi^{(n)},W)$ and $(A,g,\varepsilon,\xi,W)$ give, for every $t\in[0,T]$,

$$\mathbb{E}\big[|X^{(n)}_t-X_t|^2\big]\le\Big(\mathbb{E}\big[|\xi^{(n)}-\xi|^2\big]+\int_0^t\mathbb{E}\big[|g^{(n)}_r-g_r|^2\big]\,dr\Big)\exp(\kappa\,t)\le\rho_n,$$

where $\rho_n=\big(\mathbb{E}[|\xi^{(n)}-\xi|^2]+\int_0^T\mathbb{E}[|g^{(n)}_r-g_r|^2]\,dr\big)\exp(\kappa\,T)$: the integral over $[0,t]$ is at most the integral over $[0,T]$ — by additivity over adjacent intervals, monotonicity of the integral, and nonnegativity of the continuous integrand, the endpoint cases by the convention $\int_0^0:=0$ — and $\exp(\kappa\,t)\le\exp(\kappa\,T)$ because the exponential function is increasing, by claim 4 of the properties of the exponential function, and $\kappa\ge0$. The set $\{\mathbb{E}[|X^{(n)}_t-X_t|^2]:t\in[0,T]\}$ is nonempty and bounded above by $\rho_n$, so its supremum exists by the least-upper-bound property of the real numbers and satisfies $0\le\sup\{\mathbb{E}[|X^{(n)}_t-X_t|^2]:t\in[0,T]\}\le\rho_n$. By the hypotheses of clause (b) and the algebra of limits of real sequences, $\rho_n\to0$ as $n\to\infty$, and claim 2 (the squeeze) of the order properties of limits establishes clause (b).\ $\square$