Discretely sampled signals and the rough Hoff process

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 $X:[0,1]\to {\mathbb{R}}^d$ at a set of times $D=(t_i)$, we construct a piecewise linear, axis-directed process $X^D:[0,1]\to {\mathbb{R}}^{2d}$ comprised of a past and 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 Itô 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. Such random ODEs have a natural interpretation in the context of mathematical finance.

References in corpus (8)

• Rough paths, Signatures and the modelling of functions on streams
• Learning from the past, predicting the statistics for the future, learning an evolving system
• Non-standard approximations of the Ito-map
• Extracting information from the signature of a financial data stream
• Differential Equations Driven by Gaussian Signals II
• Rough path recursions and diffusion approximations
• The Brownian Frame Process as a Rough Path
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• Approximate Bayesian Computation with Path Signatures
• Machine-learning regression methods for American-style path-dependent contracts