Nested polygonal approximation of Brownian rough paths

For deterministic nested partitions with mesh tending to zero, lift the polygonal interpolation of Brownian motion by its ordinary iterated integrals. These path signatures converge almost surely to enhanced Brownian motion in every Hölder rough-path metric of exponent below $1/2$, hence in $p$-variation for $p>2$. Conditioning on the partition values gives a nested martingale for the first level and the off-diagonal second levels. Moment bounds and interpolation upgrade pointwise convergence to convergence in the rough path metric.

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.