= Factorial Picard estimate for a linear stochastic differential equation
{c}
{title2=$\mathbb E|D_m(t)|^2\leq |a|^2(Kt)^{m+1}/(m+1)!$}
For $dy=\sum_iA^iy\,dB^i$ with constant linear maps, let $K=\sum_i\|A^i\|_{\mathrm{op}}^2$ and let $D_m$ be consecutive <Picard iterate> differences starting from $y^{(0)}=a$. The vector <Itô isometry> gives $\mathbb E|D_m(t)|^2\leq K\int_0^t\mathbb E|D_{m-1}(s)|^2ds$, 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 $\sum_m\mathbb E\sup_{t\leq T}|D_m(t)|$ finite. <Tonelli's theorem> proves absolute uniform almost-sure convergence of the iterations, in addition to their $L^2$ convergence.
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