Brownian rough path martingale Hölder bound ($\Vert S_2(B^{(n)}){\Vert }_{\alpha }\le C\sqrt{\mathbb{E}[\Vert \mathbf{B}{\Vert }_{\alpha }^2\mid {\mathcal{F}}_n]}$)

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.