= Brownian rough path martingale Hölder bound
{c}
{title2=$\|S_2(B^{(n)})\|_\alpha\leq C\sqrt{\mathbb E[\|\mathbf B\|_\alpha^2\mid\mathcal F_n]}$}
Condition on the Brownian values at a dyadic mesh. First-level increments and antisymmetric area increments of its polygonal lift are conditional expectations of the corresponding Brownian increments. Conditional Jensen inequalities and the homogeneous degree-two norm give the displayed bound. The <Doob maximal inequality> makes the right-hand side uniformly finite almost surely when the Brownian Hölder norm has sufficiently many moments. Dense-time martingale convergence and equicontinuity then yield simultaneous convergence at every time.
Back to article page
