Discretely sampled signals and the rough Hoff process
By Guy Flint, Ben M. Hambly, Terry Lyons. 2013
47 citations recorded by the source corpus.
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]→Rd at a set of times D={ti}, we construct a piecewise linear, axis-directed process XD:[0,1]→R2d comprised of a past and a future component. We call such an object the Hoff process associated with the discrete data {Xt}ti∈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 XD. 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.
Subfields: Economics and Econometrics, Finance
View Paper: https://www.semanticscholar.org/paper/2df5f8e9cb9d24533fc3988b38d7e4c6844f3a06
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In Economics and Econometrics
In Finance
- Huilin Zhang
- Peter K. Friz
- Nicolas Victoir
- T. Cass
- Martin Hairer
- Terry Lyons
- Samy Tindel
- Archil Gulisashvili
Citations in the corpus
- Differential equations driven by rough paths with jumps (2017, methodology): “In [9] the authors consider piecewise linear, axis-directed approximations of continuous semimartingale and so obtain an Itô SDE in the Wong-Zakai limit; in [33] Itô SDEs are obtained as averaged Stratonovich solutions.”