Discretely sampled signals and the rough Hoff process

作者：Flint Guy^*; Hambly Ben; Lyons Terry

来源：Stochastic Processes and Their Applications, 2016, 126(9): 2593-2614.

DOI：10.1016/j.spa.2016.02.011

• Rough path theory
• Lead-lag path
• Hoff process
• Wong-Zakai approximations
• Ito-Stratonovich correction

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] -> R-d at a set of times D = {t(i)}, we construct a piecewise linear, axis-directed process X-D : [0, 1] -> R-2d comprised of a past and a future component. We call such an object the Hoff process associated with the discrete data {X-t}(ti is an element of 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 Ito 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 (Wong and Zakai, 1965). Such random ODEs have a natural interpretation in the context of mathematical finance.

• 出版日期2016-9