Discretely sampled signals and the rough Hoff process

We introduce a canonical method for transforming a discrete sequential data set into an associated rough path made up of lead-lag increments. In particular, by sampling a $d$-dimensional continuous semimartingale $X:[0,1] \rightarrow \mathbb{R}^d$ at a set of times $D=(t_i)$, we construct a piecewise linear, axis-directed process $X^D: [0,1] \rightarrow\mathbb{R}^{2d}$ comprised of a past and future component. We call such an object the Hoff process associated with the discrete data $\{X_{t}\}_{t_i\in D}$. The Hoff process can be lifted to its natural rough path enhancement and we consider the question of convergence as the sampling frequency increases. We prove that the It\^{o} integral can be recovered from a sequence of random ODEs driven by the components of $X^D$. This is in contrast to the usual Stratonovich integral limit suggested by the classical Wong-Zakai Theorem. Such random ODEs have a natural interpretation in the context of mathematical finance.