Discretely sampled signals and the rough Hoff process

Guy Flint, Ben Hambly and Terry Lyons

Stochastic Processes and their Applications, 2016, vol. 126, issue 9, 2593-2614

Abstract: 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 Itô 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.

Keywords: Rough path theory; Lead–lag path; Hoff process; Wong–Zakai approximations; Itô–Stratonovich correction (search for similar items in EconPapers)
Date: 2016
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Citations: View citations in EconPapers (6)

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DOI: 10.1016/j.spa.2016.02.011

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