an:06610056
Zbl 1348.60058
Flint, Guy; Hambly, Ben; Lyons, Terry
Discretely sampled signals and the rough Hoff process
EN
Stochastic Processes Appl. 126, No. 9, 2593-2614 (2016).
00357616
2016
j
60G17 60G35 60G44 60H10
rough path theory; lead-lag path; Hoff process; Wong-Zakai approximations; It??-Stratonovich correction
Summary: 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 \(X : [0, 1] \to \mathbb{R}^d\) at a set of times \(D = \{t_i \}\), we construct a piecewise linear, axis-directed process \(X^D : [0, 1] \to \mathbb{R}^{2 d}\) comprised of a past and a 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?? 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 [\textit{E. Wong} and \textit{M. Zakai}, Int. J. Eng. Sci. 3, 213--229 (1965; Zbl 0131.16401)]. Such random ODEs have a natural interpretation in the context of mathematical finance.
Zbl 0131.16401