= Solution
{c}
Set $K=\sum_i\|A^i\|_{\mathrm{op}}^2$, where the norms are <operator norms>, and $D_m=y^{(m+1)}-y^{(m)}$. Independence of the Brownian coordinates and the vector <Itô isometry> give
$$
\mathbb E|D_0(t)|^2\leq |a|^2Kt,\qquad
\mathbb E|D_m(t)|^2\leq K\int_0^t\mathbb E|D_{m-1}(s)|^2ds.
$$
Induction, as in part (b), yields the <Factorial Picard estimate for a linear stochastic differential equation>
$$
\mathbb E|D_m(t)|^2\leq |a|^2\frac{(Kt)^{m+1}}{(m+1)!}.
$$
For $t\leq T$, the <triangle inequality> in $L^2$ therefore bounds any tail of the iterates by the corresponding tail of
$$
|a|\sum_{m\geq0}\frac{(KT)^{(m+1)/2}}{\sqrt{(m+1)!}}.
$$
This numerical series converges by the <ratio test>: successive terms have ratio $\sqrt{KT/(m+2)}\to0$. Thus the iterates are Cauchy in $L^2$, uniformly in the deterministic time parameter on each bounded interval. <Completeness of Lp spaces> gives a limit $y_t$. Once its continuous adapted version is chosen in part (d), the <Itô isometry> permits passage to the limit in the integral equation: the squared norm of each integral difference is bounded by $K\int_0^T\mathbb E|y_s^{(m)}-y_s|^2ds\to0$. Consequently \b[the Picard limit solves the stochastic differential equation].
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