Discrete Hilbert Transform a la Gundy-Varopoulos

TL;DR

This paper presents a stochastic integral representation of the discrete Hilbert transform on integers, linking it to jump processes and differential subordination, inspired by continuous analogs.

Contribution

It introduces a novel stochastic representation of the discrete Hilbert transform involving jump processes, extending Gundy-Varopoulos' continuous framework to the discrete setting.

Findings

Representation of the discrete Hilbert transform as a conditional expectation of stochastic integrals.

Connection between the transform and differential subordination with quadratic covariation.

Illustration of Cauchy Riemann relations in the discrete stochastic setting.

Abstract

We show that the centered discrete Hilbert transform on integers applied to a function can be written as the conditional expectation of a transform of stochastic integrals, where the stochastic processes considered have jump components. The stochastic representation of the function and that of its Hilbert transform are under differential subordination and orthogonality relation with respect to the sharp bracket of quadratic covariation. This illustrates the Cauchy Riemann relations of analytic functions in this setting. This result is inspired by the seminal work of Gundy and Varopoulos on stochastic representation of the Hilbert transform in the continuous setting.