Discretely sampled signals and the rough Hoff path

Stochastic Analysis Seminar Series

Abstract: Sampling a $d$-dimensional continuous signal (say a semimartingale) $X:[0,T] \rightarrow \mathbb{R}^d$ at times $D=(t_i)$, we follow the recent papers [Gyurko-Lyons-Kontkowski-Field-2013] and [Lyons-Ni-Levin-2013] in constructing a lead-lag path; to be precise, a piecewise-linear, axis-directed process $X^D: [0,1] \rightarrow \mathbb{R}^{2d}$ comprised of a past and future component. Lifting $X^D$ to its natural rough path enhancement, we can consider the question of convergence as the latency of our sampling becomes finer. 

The quadratic variation of the underlying signal naturally appears as we consider the rough path limit, a phenomenon first observed by B. Hoff in his PhD thesis [Hoff-2005]. 



Monday, February 10, 2014 - 14:15
to 15:15