  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.
