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

Throughout, $\lVert\cdot\rVert_2$ and the triangle inequality are from Square-Integrable Random Variables and the Mean-Square Inner Product and Cauchy-Schwarz and Triangle Inequalities for the Mean-Square Norm; mean-square limits are unique up to almost-sure equality (claim 1 of Existence and Uniqueness of the Mean-Square Riemann Integral for Mean-Square Continuous Families for integrals, and in general immediate from the triangle inequality and the null-equivalence of Square-Integrable Random Variables and the Mean-Square Inner Product); $\alpha$ bounds all entries of $A$, $\Phi$, $\Psi$ on $[0,T]$ (Extreme Value Theorem on a Compact Interval); and matrix-entry manipulations use claims 1-3 of Componentwise Estimates, Transpose Identities, and Indefinite Riemann Integrals. All Wiener-integral linearity and interval-splitting statements used below are claim 1 of Properties of the Ito Integral: Linearity, Isometry, Martingale Property, and Mean-Square Continuity and the subinterval convention of Ito Integrable Process and the Ito Integral; all mean-square Riemann integral manipulations are claims 1, 2, 5, 6 of Basic Properties of the Mean-Square Riemann Integral.

Claim 1. Each entry $\Phi_{ij}$ satisfies $\Phi_{ij}(t)=\Phi_{ij}(0)+\int_0^t(A\Phi)_{ij}(r)\,dr$ with $(A\Phi)_{ij}$ continuous — exactly the hypothesis on the pair $(f,g)$ in Integration by Parts for Wiener Integrals and Mean-Square Riemann Integrals with $f=\Phi_{ij}$ and $g=(A\Phi)_{ij}$ (using Additivity of the Riemann Integral on Adjacent Intervals and claim 4 of Componentwise Estimates, Transpose Identities, and Indefinite Riemann Integrals for subinterval forms).

Mean-square continuity of $X$. $Y^{j}$ is mean-square continuous by claims 1 and 6 of Basic Properties of the Mean-Square Riemann Integral (finite sums of indefinite integrals); $Z^{j}$ is mean-square continuous because each summand $\int_0^t(\Psi\varepsilon)_{jj'}\,dW^{j'}$ is (the mean-square continuity statement of part 1 of Integration by Parts for Wiener Integrals and Mean-Square Riemann Integrals), with sums again by claim 1; constants and $\xi^{j}$ are trivially mean-square continuous; products with the continuous $\Phi_{ij}$ preserve mean-square continuity (claim 2 of Basic Properties of the Mean-Square Riemann Integral). Hence each $(X^{i}_t)$ is mean-square continuous, and all versions below satisfy $Y^{j}_0=Z^{j}_0=0$ by the degenerate-interval conventions of Mean-Square Riemann Integral of a Family of Random Variables and Ito Integrable Process and the Ito Integral, as required by the hypotheses of Integration by Parts for Wiener Integrals and Mean-Square Riemann Integrals.

Verification of the integral equation. Fix $t\in[0,T]$ and $i$; all identities are almost sure. We compute $\int_0^t\bigl(\sum_jA_{ij}(r)X^{j}_r+g^{i}_r\bigr)dr$ by expanding $X^{j}_r=\sum_{j''}\Phi_{jj''}(r)(\xi^{j''}+Y^{j''}_r+Z^{j''}_r)$ and using linearity (claim 1 of Basic Properties of the Mean-Square Riemann Integral) to treat three groups; note $\sum_jA_{ij}\Phi_{jj''}=(A\Phi)_{ij''}$.

Group 1 (constant random factors). By claim 2 of Basic Properties of the Mean-Square Riemann Integral and the fundamental-solution equation,

since $\Phi(0)=I_l$ (the identity matrix; degenerate interval by the convention of Mean-Square Riemann Integral of a Family of Random Variables).

Group 2 (time-integral part). For fixed $j''$, set $H_u=\sum_{j'''}\Psi_{j''j'''}(u)g^{j'''}_u$ (mean-square continuous by claims 1-2 of Basic Properties of the Mean-Square Riemann Integral); by claim 1 there, $Y^{j''}_u=\int_0^uH_v\,dv$ almost surely, the two expressions for $Y^{j''}$ agreeing at each time. Apply part 2 of Integration by Parts for Wiener Integrals and Mean-Square Riemann Integrals with $f=\Phi_{ij''}$, $g=(A\Phi)_{ij''}$:

Summing over $j''$ and using $\sum_{j''}\Phi_{ij''}(u)\Psi_{j''j'''}(u)=(\Phi\Psi)_{ij'''}(u)=\delta_{ij'''}$ (claim 2 of Fundamental Solution and Variation of Constants for Linear Ordinary Differential Equations), the first sum collapses to $\int_0^tg^{i}_u\,du$; hence

Group 3 (Wiener part). For each pair $(j'',j')$ write $V^{j''j'}_u=\int_0^u(\Psi\varepsilon)_{j''j'}\,dW^{j'}$. Part 1 of Integration by Parts for Wiener Integrals and Mean-Square Riemann Integrals with $f=\Phi_{ij''}$, $g=(A\Phi)_{ij''}$, $k=(\Psi\varepsilon)_{j''j'}$ gives

Summing over $j''$ (Wiener-integral linearity for the left side) gives, for each $j'$, the integrand $\sum_{j''}\Phi_{ij''}(\Psi\varepsilon)_{j''j'}=(\Phi\Psi\varepsilon)_{ij'}=\varepsilon_{ij'}$; then summing over $j'$,

Assembly. Adding the three groups and $\int_0^tg^{i}\,du$:

which is the equation of Mean-Square Solution of a Linear Stochastic Differential Equation with Additive Wiener Noise, rearranged.

Claim 2. Let $X,\widetilde X$ be mean-square solutions and put $v(t)=\sum_i\lVert X^{i}_t-\widetilde X^{i}_t\rVert_2$, a continuous function (claim 4 of Basic Properties of the Mean-Square Riemann Integral applied to the difference families, which are mean-square continuous). Subtracting the two defining equations (the $\xi$, $g$, and Wiener terms cancel), $X^{i}_t-\widetilde X^{i}_t=\int_0^t\sum_jA_{ij}(r)(X^{j}_r-\widetilde X^{j}_r)\,dr$ almost surely, so by the norm bound (claim 4 of Basic Properties of the Mean-Square Riemann Integral) and the entry bound $|A_{ij}|\le\alpha$,

By Gronwall's Lemma (Integral Form) (with constants $0$ and $l\alpha$; the interval is already of the form $[0,T]$), $v\le0$; since $v\ge0$, $v\equiv0$, and the null-equivalence of Square-Integrable Random Variables and the Mean-Square Inner Product gives $X^{i}_t=\widetilde X^{i}_t$ almost surely.

Claim 3. Write $\mathcal{S}$ for the closed mean-square span of the collection $\{\xi^{j}\}\cup\{g^{j}_r:r\le t\}\cup\{W^{j'}_r:r\le t\}$; by claim 1 of The Closed Mean-Square Span of a Family of Random Variables, $\mathcal{S}$ is closed under finite linear combinations and mean-square limits and contains the collection. First, $Y^{j}_t\in\mathcal{S}$: it is a finite sum of mean-square Riemann integrals over $[0,t]$, each a mean-square limit of its mean-square Riemann sums (Mean-Square Riemann Integral of a Family of Random Variables), which are finite linear combinations of values $\Psi_{jj''}(\tau)g^{j''}_\tau$ with $\tau\le t$ — scalar multiples of members of the collection. Second, $Z^{j}_t\in\mathcal{S}$: each Wiener integral $\int_0^t(\Psi\varepsilon)_{jj'}\,dW^{j'}$ is, by claim 1 of Adapted Mean-Square Continuous Processes are Ito Integrable and claim 1 of Existence and Uniqueness of the Mean-Square Extension of the Elementary Stochastic Integral, a mean-square limit of elementary stochastic integrals of simple approximants taking the deterministic grid values of the integrand; each such elementary integral is a finite linear combination of increments $W^{j'}_{t_{p+1}}-W^{j'}_{t_p}$ with $t_p\le t$, hence of members of the collection. Finally $X^{i}_t=\sum_j\Phi_{ij}(t)(\xi^{j}+Y^{j}_t+Z^{j}_t)$ is a finite linear combination of members of $\mathcal{S}$, hence lies in $\mathcal{S}$; this is claim 3. $\blacksquare$