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 -dimensional continuous semimartingale at a set of times , we construct a piecewise linear, axis-directed process comprised of...
| Main Authors: | , , |
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| פורמט: | Journal article |
| יצא לאור: |
Elsevier
2016
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| סיכום: | 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 -dimensional continuous semimartingale at a set of times , we construct a piecewise linear, axis-directed process comprised of a past and a future component. We call such an object the Hoff process associated with the discrete data . 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 Itô integral can be recovered from a sequence of random ODEs driven by the components of . 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. |
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