Factorial Picard estimate for a linear stochastic differential equation ($\mathbb{E}|D_m(t){|}^2\le |a{|}^2(Kt{)}^{m+1}/(m+1)!$)

For $dy={\sum }_i{A}^{i}yd{B}^i$ with constant linear maps, let $K={\sum }_i\Vert A^i{\Vert }_{\mathrm{op}}^2$ and let $D_m$ be consecutive Picard iterate differences starting from $y^{(0)}=a$. The vector Itô isometry gives $\mathbb{E}|D_m(t){|}^2\le K{\int }_0^t\mathbb{E}|D_{m-1}(s){|}^{2}ds$, proving the displayed estimate by induction. The square roots of these factorial bounds are summable on bounded time intervals. The Doob L2 maximal inequality then makes ${\sum }_m\mathbb{E}{\sup }_{t\le T}|D_m(t)|$ finite. Tonelli's theorem proves absolute uniform almost-sure convergence of the iterations, in addition to their $L^2$ convergence.